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Class 11 • Physics
Oscillations
Chapter-13
369 Questions
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1
MediumBITSAT2024
Five identical springs are used in the three configurations as shown in figure. The time periods of vertical oscillations in configurations (a), (b) and (c) are in the ratio.
Options:
A) 1: \sqrt{2}: \frac{1}{\sqrt{2}}
B) 2: \sqrt{2}: \frac{1}{\sqrt{2}}
C) \frac{1}{\sqrt{2}}: 2: 1
D) 2: \frac{1}{\sqrt{2}}: 1
2
MediumBITSAT2023
The $x-t graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at t=2. \mathrm{~s}$ is
Options:
A) -\frac{\pi^2}{8} \mathrm{~ms}^{-2}
B) \frac{\pi^2}{16} \mathrm{~ms}^{-2}
C) -\frac{\pi^2}{16} \mathrm{~ms}^{-2}
D) \frac{\pi^2}{8} \mathrm{~ms}^{-2}
3
MediumBITSAT2021
A tray of mass (M) 12 kg is supported by a spring as shown in the figure. When the tray is pressed down and released, it executes SHM with a period of 1.5 s. When a block of mass m placed on the tray, the period of SHM changes to 3.0 s. The mass of block is
Options:
A) 36 kg
B) 48 kg
C) 12 kg
D) 24 kg
4
MediumBITSAT2021
A particle executes SHM with a frequency f. The frequency with which its kinetic energy oscillates is
Options:
A) {f \over 2}
B) f
C) 2f
D) 4f
5
MediumBITSAT2020
A simple pendulum is placed inside a lift, the lift is moving with a uniform acceleration. If the time periods of the pendulum while the lift is moving upwards and downwards are in the ratio of 1 : 3, then the acceleration of the lift is [Take acceleration due to gravity, g = 10 m/s2]
Options:
A) 4 m/s2
B) 6 m/s2
C) 8 m/s2
D) 10 m/s2
6
MediumCOMEDK2025
Time period of oscillation of a mass suspended from a spring is T . If the spring is cut into four equal parts and the same mass is suspended from one of the parts, the fractional change in time period is
Options:
A) \frac{1}{8}
B) \frac{1}{4}
C) \frac{1}{2}
D) \frac{1}{6}
7
MediumCOMEDK2024
A body is executing SHM. When its displacements from the mean position are $4 \mathrm{~cm} and 5 \mathrm{~cm} it has velocity 10 \mathrm{~cms}^{-1} and 8 \mathrm{~cms}^{-1} respectively. Its periodic time \mathrm{t}$ is
Options:
A) \frac{2 \pi}{3} \sec
B) 2$\pi$ sec
C) \frac{3 \pi}{2} \mathrm{sec}
D) \pi$ sec
8
MediumCOMEDK2024
A circular disc of mass $20 \mathrm{~kg}, having radius 10 \mathrm{~cm} is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The time period of torsional oscillations is found to be 1 \mathrm{~s}$. The torsional spring constant of the wire is
Options:
A) 6 \mathrm{~N~m} \mathrm{~rad}^{-1}
B) 9.86 \mathrm{~N~m} \mathrm{~rad}^{-1}
C) 3.94 \mathrm{~N~m} \mathrm{~rad}^{-1}
D) 1.264 \mathrm{~N~m} \mathrm{~rad}^{-1}
9
MediumCOMEDK2024
A particle executes a simple harmonic motion of amplitude $\mathrm{A}$. The distance from the mean position at which its kinetic energy is equal to its potential energy is
Options:
A) 0.91 A
B) 0.71 A
C) 0.81 A
D) 0.51 A
10
MediumCOMEDK2023
The phase difference between displacement and velocity of a particle in simple harmonic motion is
Options:
A) \pi \mathrm{~rad}
B) 3 \pi / 2 \mathrm{~rad}
C) zero
D) \pi / 2 \mathrm{~rad}
11
MediumCOMEDK2022
A particle is performing simple harmonic motion. Equation of its motion is $x = 5\sin \left( {4t - {\pi \over 6}} \right),x being the displacement from mean position. Velocity (in ms^{-1}$) of the particle at the instant when its displacement is 3, will be
Options:
A) {{2\pi } \over 3}
B) {{5\pi } \over 6}
C) 20
D) 16
12
MediumCOMEDK2021
The function $y = \log \omega t$ can represent
Options:
A) a periodic function
B) non-periodic function
C) oscillatory function
D) circulatory motion
13
MediumCOMEDK2021
Two spring of force constant $k_1 and k_2$ are configured as the figure given below The angular frequency of this configuration is
Options:
A) {{{k_1} + {k_2}} \over m}
B) \sqrt {{{{k_1} + {k_2}} \over m}}
C) \sqrt {{{{k_1}} \over {{k_2}m}}}
D) {{{k_2}m} \over {{k_1}}}
14
MediumCOMEDK2020
A particle executes a linear SHM with an amplitude of 4 cm. At the mean position the velocity of the particle is 10 cm/s. What is the displacement of the particle when its speed becomes 5 cm/s?
Options:
A) 2($\sqrt3$) cm
B) 2($\sqrt5$) cm
C) \sqrt5$ cm
D) \sqrt3$ cm
15
MediumJee Advance2025
As shown in the figures, a uniform rod OO' of length l is hinged at the point O and held in place vertically between two walls using two massless springs of same spring constant. The springs are connected at the midpoint and at the top-end (O') of the rod, as shown in Fig. 1 and the rod is made to oscillate by a small angular displacement. The frequency of oscillation of the rod is f₁. On the other hand, if both the springs are connected at the midpoint of the rod, as shown in Fig. 2 and the rod is made to oscillate by a small angular displacement, then the frequency of oscillation is f₂. Ignoring gravity and assuming motion only in the plane of the diagram, the value of \frac{f_1}{f_2} is:
Options:
A) 2
B) \sqrt{2}
C) \sqrt{\frac{5}{2}}
D) \sqrt{\frac{2}{5}}
16
MediumJee Advance2012
A small block is connected to one end of a massless spring of un-stretched length 4.9 m. The other end of the spring (see the figure) is fixed. The system lies on a horizontal frictionless surface. The block is stretched by 0.2 m and released from rest at t = 0. It then executes simple harmonic motion with angular frequency $\omega = (\pi/3) rad/s. Simultaneously, at t = 0, a small pebble is projected with speed v from point P at an angle of 45^\circ$ as shown in the figure. Point O is at a horizontal distance of 10 m from O. If the pebble hits the block at t = 1 s, the value of v is (take g = 10 m/s2)
Options:
A) \sqrt {50} $ m/s
B) \sqrt {51} $ m/s
C) \sqrt {52} $ m/s
D) \sqrt {53} $ m/s
17
MediumJee Advance2011
Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.
Options:
A)
B)
C)
D)
18
MediumJee Advance2011
Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.
Options:
A) E1 = $\sqrt2$E2
B) E1 = 2E2
C) E1 = 4E2
D) E1 = 16E2
19
MediumJee Advance2011
Phase space diagrams are useful tools in analyzing all kinds of dynamical problems. They are especially useful in studying the changes in motion as initial position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is x(t) vs. p(t) curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown int he figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.
Options:
A)
B)
C)
D)
20
EasyJee Advance2011
A wooden block performs $SHM on a frictionless surface with frequency, {v_0}. The block carries a charge +Q on its surface . If now a uniform electric field \overrightarrow E is switched- on as shown, then the SHM$ of the block will be
Options:
A) of the same frequency and with shifted mean position.
B) of the same frequency and with the same mean position
C) of changed frequency and with shifted mean position.
D) of changed frequency and with the same mean position.
21
EasyJee Advance2011
A point mass is subjected to two simultaneous sinusoidal displacements in x-direction, ${x_1}\left( t \right) = A\sin \omega t and {x_2}\left( t \right) = A\sin \left( {\omega t + {{2\pi } \over 3}} \right). Adding a third sinusoidal displacement {x_3}\left( t \right) = B\sin \left( {\omega t + \phi } \right) brings the mass to a complete rest. The values of B and \phi $ are
Options:
A) \sqrt 2 A,{{3\pi } \over 4}
B) A,{{4\pi } \over 3}
C) \sqrt 3 A,{{5\pi } \over 6}
D) A,{\pi \over 3}
22
EasyJee Advance2010
When a particle of mass m moves on the x-axis in a potential of the form V(x) = kx2, it performs simple harmonic motion. The corresponding time period is proportional to $\sqrt {{m \over k}} , as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx2 and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is V(x) = \alphax4 (\alpha$ > 0) for | x | near the origin and becomes a constant equal to V0 for (see figure).
Options:
A) E < 0
B) E > 0
C) V0 > E > 0
D) E > V0
23
MediumJee Advance2010
When a particle of mass m moves on the x-axis in a potential of the form V(x) = kx2, it performs simple harmonic motion. The corresponding time period is proportional to $\sqrt {{m \over k}} , as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx2 and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is V(x) = \alphax4 (\alpha$ > 0) for | x | near the origin and becomes a constant equal to V0 for (see figure).
Options:
A) A\sqrt {m/\alpha }
B) {1 \over A}\sqrt {m/\alpha }
C) A\sqrt {\alpha /m}
D) {1 \over A}\sqrt {\alpha /m}
24
EasyJee Advance2010
When a particle of mass m moves on the x-axis in a potential of the form V(x) = kx2, it performs simple harmonic motion. The corresponding time period is proportional to $\sqrt {{m \over k}} , as can be seen easily using dimensional analysis. However, the motion of a particle can be periodic even when its potential energy increases on both sides of x = 0 in a way different from kx2 and its total energy is such that the particle does not escape to infinity. Consider a particle of mass m moving on the x-axis. Its potential energy is V(x) = \alphax4 (\alpha$ > 0) for | x | near the origin and becomes a constant equal to V0 for (see figure).
Options:
A) proportional to V0.
B) proportional to V0/mX0.
C) proportional to $\sqrt {{V_0}/m{X_0}} $.
D) zero.
25
MediumJee Advance2009
The mass M shown in the figure below oscillates in simple harmonic motion with amplitude A. The amplitude of the point P is
Options:
A) {{{k_1}A} \over {{k_2}}}
B) {{{k_2}A} \over {{k_1}}}
C) {{{k_1}A} \over {{k_1} + {k_2}}}
D) {{{k_2}A} \over {{k_1} + {k_2}}}
26
MediumJee Advance2009
A uniform rod of length L and mass M is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $k. The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle \theta$ in one direction and released. The frequency of oscillation is
Options:
A) {1 \over {2\pi }}\sqrt {{{2k} \over M}}
B) {1 \over {2\pi }}\sqrt {{k \over M}}
C) {1 \over {2\pi }}\sqrt {{{6k} \over M}}
D) {1 \over {2\pi }}\sqrt {{{24k} \over M}}
27
MediumJee Advance2009
The $x-t graph of a particle undergoing simple harmonic motion is shown in the figure. The acceleration of the particle at t=4/3$ s is
Options:
A) {{\sqrt 3 } \over {32}}{\pi ^2} cm/s^2
B) {{ - {\pi ^2}} \over {32}} cm/s^2
C) {{ {\pi ^2}} \over {32}} cm/s^2
D) - {{\sqrt 3 } \over {32}}{\pi ^2} cm/s^2
28
MediumJee Advance2008
Column I gives a list of possible set of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column II. Match the set of parameters given in Column I with the graphs given in Column II. Indicate your answer by darkening the appropriate bubbles of the 4 $\times$ 4 matrix given in the ORS. Column I Column II (A) Potential energy of a simple pendulum (y-axis) as a function of displacement (x) axis (P) (B) Displacement (y-axis) as a function of time (x-axis) for a one dimensional motion at zero or constant acceleration when the body is moving along the positive x-direction (Q) (C) Range of a projectile (y-axis) as a function of its velocity (x-axis) when projected at a fixed angle (R) (D) The square of the time period (y-axis) of a simple pendulum as a function of its length (x-axis) (S)
Options:
A) A$\to(P, S); B\to(Q, S); C\to(S); (D)\to$(Q)
B) A$\to(S); B\to(Q, S); C\to(S); (D)\to$(Q, S)
C) A$\to(P, S); B\to(Q); C\to(S); (D)\to$(Q, S)
D) A$\to(S); B\to(Q, S); C\to(S, P); (D)\to$(Q)
29
MediumJee Advance2005
A simple pendulum has time period T1. The point of suspension is now moved upward according to the relation y = Kt2, (K = 1 m/s2) where y is the vertical displacement. The time period now become T2. The ratio of ${{T_1^2} \over {T_2^2}}$ is (g = 10 m/s2)
Options:
A) {5 \over 6}
B) {6 \over 5}
C) 1
D) {4 \over 5}
30
MediumJee Advance2005
A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency $\omega and amplitude a. If at a height y^{\prime} from the mean position, the body gets detached from the spring, calculate the value of y^{\prime} so that the height \mathrm{H} attained by the mass is maximum. The body does not interact with the spring during its subsequent motion after detachment \left(a \omega^{2}>g\right)
Options:
A) y=\frac{g}{\omega^{2}}
B) y=\frac{2g}{\omega^{2}}
C) y=\frac{g}{3\omega^{2}}
D) y=\frac{4g}{7\omega^{2}}
31
MediumJee Advance2001
A particle executes simple harmonic motion between x = - A to x = + A. The time taken for it to go from 0 to ${A \over 2} is T1 and to go from {A \over 2}$ to A is T2. Then
Options:
A) T1 < T2
B) T1 > T2
C) T1 = T2
D) T1 = 2T2
32
MediumJee Advance2000
The period of oscillation of a simple pendulum of length $L suspended from the roof of a vehicle which moves without friction down an inclined plane of inclination \alpha$, is given by
Options:
A) 2\pi \sqrt {{L \over {g\cos \alpha }}}
B) 2\pi \sqrt {{L \over {g\sin \alpha }}}
C) 2\pi \sqrt {{L \over g}}
D) 2\pi \sqrt {{L \over {g\tan \alpha }}}
33
MediumJee AdvanceIIT-JEE 1999 Screening
A particle free to move along the x-axis has potential energy given by $U\left( x \right) = k\left[ {1 - \exp \left( { - {x^2}} \right)} \right] for - \infty \le x \le - \infty $, where k is a positive constant of appropriate dimensions. Then
Options:
A) at points away from the origin, the particle is in unstable equilibrium
B) for any finite nonzero value of x, there is a force directed away from the origin
C) if its total mechanical energy is ${k \over 2}$, it has its minimum kinetic energy at the origin
D) for small displacement from x = 0, the motion is simple harmonic
34
MediumJee AdvanceIIT-JEE 1988
Two bodies M and N of equal masses are suspended from two separate massless springs of spring constant k1 and k2 respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of vibration of M to that of N is
Options:
A) {{{k_1}} \over {{k_2}}}
B) \sqrt {{{{k_1}} \over {{k_2}}}}
C) {{{k_2}} \over {{k_1}}}
D) \sqrt {{{{k_2}} \over {{k_1}}}}
35
MediumJee Advance2025
A person sitting inside an elevator performs a weighing experiment with an object of mass 50 kg . Suppose that the variation of the height y (in m ) of the elevator, from the ground, with time t (in s) is given by y=8\left[1+\sin \left(\frac{2 \pi t}{T}\right)\right], where T=40 \pi \mathrm{~s}. Taking acceleration due to gravity, g=10 \mathrm{m} / \mathrm{s}^2, the maximum variation of the object's weight (in N ) as observed in the experiment is ___________.
Options:
36
HardJee Advance2024
Two particles, 1 and 2, each of mass m, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at x_0, are oscillating with amplitude a and angular frequency \omega. Thus, their positions at time t are given by x_1(t)=\left(x_0+d\right)+a \sin \omega t and x_2(t)=\left(x_0-d\right)-a \sin \omega t, respectively, where d>2 a. Particle 3 of mass m moves towards this system with speed u_0=a \omega / 2, and undergoes instantaneous elastic collision with particle 2 , at time t_0. Finally, particles 1 and 2 acquire a center of mass speed v_{\mathrm{cm}} and oscillate with amplitude b and the same angular frequency \omega.
Options:
37
HardJee Advance2024
Two particles, 1 and 2, each of mass m, are connected by a massless spring, and are on a horizontal frictionless plane, as shown in the figure. Initially, the two particles, with their center of mass at x_0, are oscillating with amplitude a and angular frequency \omega. Thus, their positions at time t are given by x_1(t)=\left(x_0+d\right)+a \sin \omega t and x_2(t)=\left(x_0-d\right)-a \sin \omega t, respectively, where d>2 a. Particle 3 of mass m moves towards this system with speed u_0=a \omega / 2, and undergoes instantaneous elastic collision with particle 2 , at time t_0. Finally, particles 1 and 2 acquire a center of mass speed v_{\mathrm{cm}} and oscillate with amplitude b and the same angular frequency \omega.
Options:
38
HardJee Advance2022
A particle of mass 1 \mathrm{~kg} is subjected to a force which depends on the position as \vec{F}= -k(x \hat{\imath}+y \hat{\jmath}) \mathrm{kg}\, \mathrm{m} \mathrm{s}^{-2} with k=1 \mathrm{~kg} \mathrm{~s}^{-2}. At time t=0, the particle's position \vec{r}= \left(\frac{1}{\sqrt{2}} \hat{\imath}+\sqrt{2} \hat{\jmath}\right) m and its velocity \vec{v}=\left(-\sqrt{2} \hat{\imath}+\sqrt{2} \hat{\jmath}+\frac{2}{\pi} \hat{k}\right) m s^{-1}. Let v_{x} and v_{y} denote the x and the y components of the particle's velocity, respectively. Ignore gravity. When z=0.5 \mathrm{~m}, the value of \left(x v_{y}-y v_{x}\right) is __________ m^{2} s^{-1}.
Options:
39
MediumJee Advance2022
On a frictionless horizontal plane, a bob of mass m=0.1 \mathrm{~kg} is attached to a spring with natural length l_{0}=0.1 \mathrm{~m}. The spring constant is k_{1}=0.009 \,\mathrm{Nm}^{-1} when the length of the spring l>l_{0} and is k_{2}=0.016 \,\mathrm{Nm}^{-1} when l < l_{0}. Initially the bob is released from l= 0.15 \mathrm{~m}. Assume that Hooke's law remains valid throughout the motion. If the time period of the full oscillation is T=(n \pi) s, then the integer closest to n is __________.
Options:
40
MediumJee Advance2010
A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1 m and its crosssectional area is 4.9 $ \times 10-7 m2. If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency 140 rad s−1. If the Young’s modulus of the material of the wire is n \times $ 109 Nm-2, the value of n is
Options:
41
MediumJee AdvanceIIT-JEE 1994
An object of mass 0.2 kg executes simple harmonic oscillation along the x-axis with a frequency of $\left( {{{25} \over \pi }} \right)$ Hz. At the position x = 0.04, the object has kinetic energy of 0.5 J and potential energy 0.4 J. The amplitude of oscillations is ................ m.
Options:
42
HardJee Advance2016
A block with mass M is connected by a massless spring with stiffness constant k to a rigid wall and moves without friction on a horizontal surface. The block oscillates with small amplitude A about an equilibrium position x0. Consider two cases: (i) when the block is at x0; and (ii) when the block is at x = x0 + A. In both cases, a particle with mass m( < M) is softly placed on the block after which they stick on each other. Which of the following statement(s) is(are) true about the motion after the mass m is placed on the mass M?
Options:
A) The amplitude of oscillation in the first case changes by a factor of $\sqrt {{M \over {m + M}}} $, whereas in the second case it remains unchanged.
B) The final time period of oscillation in both the cases is same
C) The total energy decreases in both the cases
D) The instantaneous speed at x0 of the combined masses decreases in both the cases
43
HardJee Advance2015
Two independent harmonic oscillators of equal masses are oscillating about the origin with angular frequencies $\omega1 and \omega2 and have total energies E1 and E2, respectively. The variations of their momenta p with positions x are shown in the figures. If {a \over b} = {n^2} and {a \over R} = n$, then the correct equations is/are
Options:
A) E1$\omega1 = E2\omega$2
B) {{{\omega _2}} \over {{\omega _1}}} = {n^2}
C) {\omega _1}{\omega _2} = {n^2}
D) {{{E_1}} \over {{\omega _1}}} = {{{E_2}} \over {{\omega _2}}}
44
EasyJee Advance2009
A student performed the experiment to measure the speed of sound in air using resonance air-column method. Two resonances in the air-column were obtained by lowering the water level. The resonance with the shorter air-column is the first resonance and that with the longer air-column is the second resonance. Then,
Options:
A) the intensity of the sound heard at the first resonance was more than that at the second resonance.
B) the prongs of the tuning fork were kept in a horizontal plane above the resonance tube.
C) the amplitude of vibration of the ends of the prongs is typically around 1 cm.
D) the length of the air-column at the first resonance was somewhat shorter than 1/4th of the wavelength of the sound in air.
45
MediumJee Advance2006
Function x=\mathrm{A} \sin ^2 \omega t+\mathrm{B} \cos ^2 \omega t+\mathrm{C} \sin \omega t \cos \omega t represents SHM
Options:
A) for any value ol \mathrm{A}, \mathrm{B} and C (except \mathrm{C}=0 ).
B) if \mathrm{A}=-\mathrm{B} ; \mathrm{C}=2 \mathrm{~B}, amplitude =|\mathrm{B} \sqrt{2}|.
C) if \mathrm{A}=\mathrm{B} ; \mathrm{C}=0.
D) if \mathrm{A}=\mathrm{B} ; \mathrm{C}=2 \mathrm{~B}, amplitude =|\mathrm{B}|
46
MediumJEE Mains2026
As shown in the figure, a spring is kept in a stretched position with some extension by holding the masses 1 \text{ kg} and 0.2 \text{ kg} with a separation more than spring natural length and are released. Assuming the horizontal surface to be frictionless, the angular frequency (in SI unit) of the system is :
Options:
A) 20
B) 5
C) 30
D) 27
47
MediumJEE Mains2026
A cylindrical block of mass M and area of cross section A is floating in a liquid of density \rho and with its axis vertical. When depressed a little and released the block starts oscillating. The period of oscillation is \_\_\_\_
Options:
A) 2 \pi \sqrt{\frac{\rho A}{M g}}
B) \pi \sqrt{\frac{2 M}{\rho A g}}
C) 2 \pi \sqrt{\frac{M}{\rho A g}}
D) \pi \sqrt{\frac{\rho A}{M g}}
48
EasyJEE Mains2026
A spring of force constant 15 \mathrm{~N} / \mathrm{m} is cut into two pieces. If the ratio of their length is 1: 3, then the force constant of smaller piece is \_\_\_\_ \mathrm{N} / \mathrm{m}.
Options:
A) 20
B) 45
C) 60
D) 15
49
EasyJEE Mains2026
A simple pendulum of string length 30 cm performs 20 oscillations in 10 s . The length of the string required for the pendulum to perform 40 oscillations in the same time duration is \_\_\_\_ cm . [Assume that the mass of the pendulum remains same.]
Options:
A) 0.75
B) 7.5
C) 15
D) 120
50
EasyJEE Mains2026
Using a simple pendulum experiment g is determind by measuring its time period T. Which of the following plots represent the correct relation between the pendulum length L and time period T ?
Options:
A)
B)
C)
D)
51
MediumJEE Mains2026
The kinetic energy of a simple harmonic oscillator is oscillating with angular frequency of 176 rad/s. The frequency of this simple harmonic oscillator is ______ Hz. [ take \pi = \frac{22}{7} ]
Options:
A) 88
B) 14
C) 28
D) 176
52
EasyJEE Mains2025
A block of mass 2 kg is attached to one end of a massless spring whose other end is fixed at a wall. The spring-mass system moves on a frictionless horizontal table. The spring's natural length is 2 m and spring constant is 200 N/m. The block is pushed such that the length of the spring becomes 1 m and then released. At distance x m (x < 2) from the wall, the speed of the block will be
Options:
A) 10\left[1-(2-x)^2\right]^{\frac{1}{2}} \ m/s
B) 10\left[1-(2-x)^2\right]^{\frac{3}{2}} \ m/s
C) 10\left[1-(2-x)^2\right] \ m/s
D) 10\left[1-(2-x)^2\right]^2 \ m/s
53
EasyJEE Mains2025
Two simple pendulums having lengths l_1 and l_2 with negligible string mass undergo angular displacements \theta_1 and \theta_2, from their mean positions, respectively. If the angular accelerations of both pendulums are same, then which expression is correct?
Options:
A) \theta_1 l_2=\theta_2 l_1
B) \theta_1 l_1=\theta_2 l_2
C) \theta_1 l_2^2=\theta_2 l_1^2
D) \theta_1 l_1^2=\theta_2 l_2^2
54
MediumJEE Mains2025
Two blocks of masses m and M,(M>m), are placed on a frictionless table as shown in figure. A massless spring with spring constant k is attached with the lower block. If the system is slightly displaced and released, then ( \mu= coefficient of friction between the two blocks) A. The time period of small oscillation of the two blocks is T=2 \pi \sqrt{\frac{(m+M)}{k}} B. The acceleration of the blocks is a=-\frac{k x}{M+m} ( x= displacement of the blocks from the mean position) C. The magnitude of the frictional force on the upper block is \frac{m \mu|x|}{M+m} D. The maximum amplitude of the upper block, if it does not slip, is \frac{\mu(M+m) g}{k} E. Maximum frictional force can be \mu(\mathrm{M}+\mathrm{m}) \mathrm{g}. Choose the correct answer from the options given below :
Options:
A) B, C, D Only
B) C, D, E Only
C) A, B, D Only
D) A, B, C Only
55
MediumJEE Mains2025
A particle is subjected to two simple harmonic motions as : $ x_1=\sqrt{7} \sin 5 \mathrm{tcm} and x_2=2 \sqrt{7} \sin \left(5 t+\frac{\pi}{3}\right) \mathrm{cm} where x is displacement and t is time in seconds. The maximum acceleration of the particle is x \times 10^{-2} \mathrm{~ms}^{-2}. The value of x$ is :
Options:
A) 5 \sqrt{7}
B) 125
C) 25 \sqrt{7}
D) 175
56
EasyJEE Mains2025
Two bodies A and B of equal mass are suspended from two massless springs of spring constant k1 and k2, respectively. If the bodies oscillate vertically such that their amplitudes are equal, the ratio of the maximum velocity of A to the maximum velocity of B is
Options:
A) \sqrt{\frac{k_2}{k_1}}
B) \sqrt{\frac{k_1}{k_2}}
C) \frac{k_2}{k_1}
D) \frac{k_1}{k_2}
57
EasyJEE Mains2025
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Time period of a simple pendulum is longer at the top of a mountain than that at the base of the mountain. Reason (R) : Time period of a simple pendulum decreases with increasing value of acceleration due to gravity and vice-versa. In the light of the above statements, choose the most appropriate answer from the options given below :
Options:
A) Both (A) and (R) are true but (R) is not the correct explanation of (A).
B) (A) is true but (R) is false.
C) Both (A) and (R) are true and (R) is the correct explanation of (A).
D) (A) is false but (R) is true.
58
MediumJEE Mains2025
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : Knowing initial position \mathrm{x}_0 and initial momentum p_0 is enough to determine the position and momentum at any time t for a simple harmonic motion with a given angular frequency \omega. Reason (R) : The amplitude and phase can be expressed in terms of \mathrm{X}_0 an \mathrm{p}_0. In the light of the above statements, choose the correct answer from the options given below :
Options:
A) (A) is true but (R) is false
B) Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
C) (A) is false but (R) is true
D) Both (A) and (R) are true and (R) is the correct explanation of (A)
59
MediumJEE Mains2025
A particle oscillates along the x-axis according to the law, x(\mathrm{t})=x_0 \sin ^2\left(\frac{\mathrm{t}}{2}\right) where x_0=1 \mathrm{~m}. The kinetic energy (\mathrm{K}) of the particle as a function of x is correctly represented by the graph
Options:
A)
B)
C)
D)
60
MediumJEE Mains2025
A particle is executing simple harmonic motion with time period 2 s and amplitude 1 cm . If D and d are the total distance and displacement covered by the particle in 12.5 s , then \frac{\mathrm{D}}{\mathrm{d}} is
Options:
A) 10
B) \frac{16}{5}
C) 25
D) \frac{15}{4}
61
MediumJEE Mains2025
A light hollow cube of side length 10 cm and mass 10 g , is floating in water. It is pushed down and released to execute simple harmonic oscillations. The time period of oscillations is y \pi \times 10^{-2} \mathrm{~s}, where the value of y is (Acceleration due to gravity, g=10 \mathrm{~m} / \mathrm{s}^2, density of water =10^3 \mathrm{~kg} / \mathrm{m}^3 )
Options:
A) 2
B) 4
C) 1
D) 6
62
MediumJEE Mains2025
Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : A simple pendulum is taken to a planet of mass and radius, 4 times and 2 times, respectively, than the Earth. The time period of the pendulum remains same on earth and the planet. Reason (R) : The mass of the pendulum remains unchanged at Earth and the other planet. In the light of the above statements, choose the correct answer from the options given below :
Options:
A) Both (A) and (R) are true and (R) is the correct explanation of (A)
B) (A) is false but (R) is true
C) (A) is true but (R) is false
D) Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
63
MediumJEE Mains2024
A simple pendulum doing small oscillations at a place $R height above earth surface has time period of T_1=4 \mathrm{~s}. \mathrm{T}_2 would be it's time period if it is brought to a point which is at a height 2 \mathrm{R} from earth surface. Choose the correct relation [\mathrm{R}=$ radius of earth] :
Options:
A) 3 \mathrm{~T}_1=2 \mathrm{~T}_2
B) \mathrm{T}_1=\mathrm{T}_2
C) 2 \mathrm{~T}_1=3 \mathrm{~T}_2
D) 2 \mathrm{~T}_1=\mathrm{T}_2
64
EasyJEE Mains2024
In simple harmonic motion, the total mechanical energy of given system is $E. If mass of oscillating particle P$ is doubled then the new energy of the system for same amplitude is:
Options:
A) E / \sqrt{2}
B) 2 E
C) E \sqrt{2}
D) E
65
MediumJEE Mains2024
A simple pendulum of length 1 \mathrm{~m} has a wooden bob of mass 1 \mathrm{~kg}. It is struck by a bullet of mass 10^{-2} \mathrm{~kg} moving with a speed of 2 \times 10^2 \mathrm{~ms}^{-1}. The bullet gets embedded into the bob. The height to which the bob rises before swinging back is. (use \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 )
Options:
A) 0.20 \mathrm{~m}
B) 0.40 \mathrm{~m}
C) 0.30 \mathrm{~m}
D) 0.35 \mathrm{~m}
66
EasyJEE Mains2024
The bob of a pendulum was released from a horizontal position. The length of the pendulum is $10 \mathrm{~m}. If it dissipates 10 \% of its initial energy against air resistance, the speed with which the bob arrives at the lowest point is: [Use, \mathrm{g}: 10 \mathrm{~ms}^{-2}$]
Options:
A) 5 \sqrt{6} \mathrm{~ms}^{-1}
B) 5 \sqrt{5} \mathrm{~ms}^{-1}
C) 2 \sqrt{5} \mathrm{~ms}^{-1}
D) 6 \sqrt{5} \mathrm{~ms}^{-1}
67
EasyJEE Mains2023
In a linear Simple Harmonic Motion (SHM) (A) Restoring force is directly proportional to the displacement. (B) The acceleration and displacement are opposite in direction. (C) The velocity is maximum at mean position. (D) The acceleration is minimum at extreme points. Choose the correct answer from the options given below:
Options:
A) {\text {(A), (B) and (D) only }}
B) (C) and (D) only
C) (A), (B) and (C) Only
D) (A), (C) and (D) only
68
EasyJEE Mains2023
A particle executes SHM of amplitude A. The distance from the mean position when its's kinetic energy becomes equal to its potential energy is :
Options:
A) \frac{1}{\sqrt{2}} A
B) \frac{1}{2} A
C) 2 \mathrm{~A}
D) \sqrt{2 A}
69
EasyJEE Mains2023
Which graph represents the difference between total energy and potential energy of a particle executing SHM vs it's distance from mean position ?
Options:
A)
B)
C)
D)
70
EasyJEE Mains2023
A particle is executing simple harmonic motion (SHM). The ratio of potential energy and kinetic energy of the particle when its displacement is half of its amplitude will be
Options:
A) 1 : 1
B) 1 : 4
C) 2 : 1
D) 1 : 3
71
EasyJEE Mains2023
The variation of kinetic energy (KE) of a particle executing simple harmonic motion with the displacement $(x)$ starting from mean position to extreme position (A) is given by
Options:
A)
B)
C)
D)
72
EasyJEE Mains2023
A particle executes S.H.M. of amplitude A along x-axis. At t = 0, the position of the particle is $x=\frac{A}{2} and it moves along positive x-axis. The displacement of particle in time t is x = A\sin (wt + \delta ), then the value of \delta$ will be
Options:
A) \frac{\pi}{2}
B) \frac{\pi}{3}
C) \frac{\pi}{4}
D) \frac{\pi}{6}
73
EasyJEE Mains2023
For particle P revolving round the centre O with radius of circular path $\mathrm{r} and angular velocity \omega, as shown in below figure, the projection of OP on the x-axis at time t$ is
Options:
A) x(t)=\operatorname{cos}\left(\omega t-\frac{\pi}{6} \omega\right)
B) x(t)=\operatorname{cos}(\omega t)
C) x(t)=r \cos \left(\omega t+\frac{\pi}{6}\right)
D) x(t)=r \sin \left(\omega t+\frac{\pi}{6}\right)
74
EasyJEE Mains2023
A mass $m is attached to two strings as shown in figure. The spring constants of two springs are \mathrm{K}_{1} and \mathrm{K}_{2}. For the frictionless surface, the time period of oscillation of mass m$ is :
Options:
A) 2\pi \sqrt {{m \over {{K_1} + {K_2}}}}
B) 2\pi \sqrt {{m \over {{K_1} - {K_2}}}}
C) {1 \over {2\pi }}\sqrt {{{{K_1} + {K_2}} \over m}}
D) {1 \over {2\pi }}\sqrt {{{{K_1} - {K_2}} \over m}}
75
EasyJEE Mains2023
Choose the correct length (L) versus square of the time period ($\mathrm{T}^{2}$) graph for a simple pendulum executing simple harmonic motion.
Options:
A)
B)
C)
D)
76
EasyJEE Mains2023
The maximum potential energy of a block executing simple harmonic motion is $25 \mathrm{~J}. A is amplitude of oscillation. At \mathrm{A / 2}$, the kinetic energy of the block is
Options:
A) 9.75 J
B) 37.5 J
C) 18.75 J
D) 12.5 J
77
EasyJEE Mains2023
For a simple harmonic motion in a mass spring system shown, the surface is frictionless. When the mass of the block is 1 \mathrm{~kg}, the angular frequency is \omega_{1}. When the mass block is 2 \mathrm{~kg} the angular frequency is \omega_{2}. The ratio \omega_{2} / \omega_{1} is
Options:
A) 1 / \sqrt{2}
B) 1 / 2
C) 2
D) \sqrt{2}
78
MediumJEE Mains2023
A particle executes simple harmonic motion between $x=-A and x=+A. If time taken by particle to go from x=0 to \frac{A}{2} is 2 s; then time taken by particle in going from x=\frac{A}{2}$ to A is
Options:
A) 4 s
B) 1.5 s
C) 3 s
D) 2 s
79
EasyJEE Mains2023
T is the time period of simple pendulum on the earth's surface. Its time period becomes $x T when taken to a height R (equal to earth's radius) above the earth's surface. Then, the value of x$ will be :
Options:
A) 4
B) \frac{1}{2}
C) 2
D) \frac{1}{4}
80
MediumJEE Mains2022
The time period of oscillation of a simple pendulum of length L suspended from the roof of a vehicle, which moves without friction down an inclined plane of inclination $\alpha$, is given by :
Options:
A) 2 \pi \sqrt{\mathrm{L} /(\mathrm{g} \cos \alpha)}
B) 2 \pi \sqrt{\mathrm{L} /(\mathrm{g} \sin \alpha)}
C) 2 \pi \sqrt{\mathrm{L} / \mathrm{g}}
D) 2 \pi \sqrt{\mathrm{L} /(\mathrm{g} \tan \alpha)}
81
EasyJEE Mains2022
Assume there are two identical simple pendulum clocks. Clock - 1 is placed on the earth and Clock - 2 is placed on a space station located at a height h above the earth surface. Clock - 1 and Clock - 2 operate at time periods 4 s and 6 s respectively. Then the value of h is - (consider radius of earth $R_{E}=6400 \mathrm{~km} and \mathrm{g} on earth 10 \mathrm{~m} / \mathrm{s}^{2}$ )
Options:
A) 1200 km
B) 1600 km
C) 3200 km
D) 4800 km
82
EasyJEE Mains2022
When a particle executes Simple Hormonic Motion, the nature of graph of velocity as a function of displacement will be :
Options:
A) Circular
B) Elliptical
C) Sinusoidal
D) Straight line
83
MediumJEE Mains2022
In figure $(\mathrm{A}), mass '2 \mathrm{~m}^{\text {' }} is fixed on mass '\mathrm{m} ' which is attached to two springs of spring constant \mathrm{k}. In figure (B), mass '\mathrm{m}' is attached to two springs of spring constant '\mathrm{k}' and '2 \mathrm{k}^{\prime}. If mass '\mathrm{m}' in (A) and in (B) are displaced by distance 'x^{\prime} horizontally and then released, then time period \mathrm{T}_{1} and \mathrm{T}_{2} corresponding to (\mathrm{A})$ and (B) respectively follow the relation.
Options:
A)
\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\frac{3}{\sqrt{2}}
B)
\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\sqrt{\frac{3}{2}}
C)
\frac{\mathrm{T}_{1}}{\mathrm{~T}_{2}}=\sqrt{\frac{2}{3}}
D)
\frac{T_{1}}{T_{2}}=\frac{\sqrt{2}}{3}
84
EasyJEE Mains2022
The motion of a simple pendulum executing S.H.M. is represented by the following equation. $y = A\sin (\pi t + \phi )$, where time is measured in second. The length of pendulum is
Options:
A) 97.23 cm
B) 25.3 cm
C) 99.4 cm
D) 406.1 cm
85
EasyJEE Mains2022
Motion of a particle in x-y plane is described by a set of following equations $x = 4\sin \left( {{\pi \over 2} - \omega t} \right)\,m and y = 4\sin (\omega t)\,m$. The path of the particle will be :
Options:
A) circular
B) helical
C) parabolic
D) elliptical
86
EasyJEE Mains2022
The equation of a particle executing simple harmonic motion is given by $x = \sin \pi \left( {t + {1 \over 3}} \right)m. At t = 1s, the speed of particle will be (Given : \pi$ = 3.14)
Options:
A) 0 cm s$-$1
B) 157 cm s$-$1
C) 272 cm s$-$1
D) 314 cm s$-$1
87
EasyJEE Mains2022
The displacement of simple harmonic oscillator after 3 seconds starting from its mean position is equal to half of its amplitude. The time period of harmonic motion is :
Options:
A) 6 s
B) 8 s
C) 12 s
D) 36 s
88
EasyJEE Mains2022
Time period of a simple pendulum in a stationary lift is 'T'. If the lift accelerates with ${g \over 6}$ vertically upwards then the time period will be : (Where g = acceleration due to gravity)
Options:
A) \sqrt {{6 \over 5}} T
B) \sqrt {{5 \over 6}} T
C) \sqrt {{6 \over 7}} T
D) \sqrt {{7 \over 6}} T
89
EasyJEE Mains2022
Two massless springs with spring constants 2 k and 9 k, carry 50 g and 100 g masses at their free ends. These two masses oscillate vertically such that their maximum velocities are equal. Then, the ratio of their respective amplitudes will be :
Options:
A) 1 : 2
B) 3 : 2
C) 3 : 1
D) 2 : 3
90
MediumJEE Mains2021
A mass of 5 kg is connected to a spring. The potential energy curve of the simple harmonic motion executed by the system is shown in the figure. A simple pendulum of length 4 m has the same period of oscillation as the spring system. What is the value of acceleration due to gravity on the planet where these experiments are performed?
Options:
A) 10 m/s2
B) 5 m/s2
C) 4 m/s2
D) 9.8 m/s2
91
MediumJEE Mains2021
A bob of mass 'm' suspended by a thread of length l undergoes simple harmonic oscillations with time period T. If the bob is immersed in a liquid that has density ${1 \over 4}$ times that of the bob and the length of the thread is increased by 1/3rd of the original length, then the time period of the simple harmonic oscillations will be :-
Options:
A) T
B) {3 \over 2}$T
C) {3 \over 4}$T
D) {4 \over 3}$T
92
MediumJEE Mains2021
For a body executing S.H.M. :(1) Potential energy is always equal to its K.E.(2) Average potential and kinetic energy over any given time interval are always equal.(3) Sum of the kinetic and potential energy at any point of time is constant.(4) Average K.E. in one time period is equal to average potential energy in one time period.Choose the most appropriate option from the options given below :
Options:
A) (3) and (4)
B) only (3)
C) (2) and (3)
D) only (2)
93
MediumJEE Mains2021
The variation of displacement with time of a particle executing free simple harmonic motion is shown in the figure.The potential energy U(x) versus time (t) plot of the particle is correctly shown in figure :
Options:
A)
B)
C)
D)
94
EasyJEE Mains2021
An object of mass 0.5 kg is executing simple harmonic motion. It amplitude is 5 cm and time period (T) is 0.2 s. What will be the potential energy of the object at an instant $t = {T \over 4}s$ starting from mean position. Assume that the initial phase of the oscillation is zero.
Options:
A) 0.62 J
B) 6.2 $\times 10-$3 J
C) 1.2 $\times$ 103 J
D) 6.2 $\times$ 103 J
95
EasyJEE Mains2021
A particle starts executing simple harmonic motion (SHM) of amplitude 'a' and total energy E. At any instant, its kinetic energy is ${{3E} \over 4}$ then its displacement 'y' is given by :
Options:
A) y = a
B) y = {a \over {\sqrt 2 }}
C) y = {{a\sqrt 3 } \over 2}
D) y = {a \over 2}
96
EasyJEE Mains2021
In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.
Options:
A) {1 \over 2}
B) {3 \over 4}
C) {1 \over 3}
D) {1 \over 4}
97
EasyJEE Mains2021
T0 is the time period of a simple pendulum at a place. if the length of the pendulum is reduced to ${1 \over {16}}$ times of its initial value, the modified time period is :
Options:
A) 4 T0
B) {1 \over {4}}$ T0
C) T0
D) 8$\pi$ T0
98
MediumJEE Mains2021
A particle is making simple harmonic motion along the X-axis. If at a distances x1 and x2 from the mean position the velocities of the particle are v1 and v2 respectively. The time period of its oscillation is given as :
Options:
A) T = 2\pi \sqrt {{{x_2^2 + x_1^2} \over {v_1^2 - v_2^2}}}
B) T = 2\pi \sqrt {{{x_2^2 + x_1^2} \over {v_1^2 + v_2^2}}}
C) T = 2\pi \sqrt {{{x_2^2 - x_1^2} \over {v_1^2 + v_2^2}}}
D) T = 2\pi \sqrt {{{x_2^2 - x_1^2} \over {v_1^2 - v_2^2}}}
99
EasyJEE Mains2021
The function of time representing a simple harmonic motion with a period of ${\pi \over \omega }$ is :
Options:
A) cos($\omegat) + cos(2\omegat) + cos(3\omega$t)
B) sin2($\omega$t)
C) sin($\omegat) + cos(\omega$t)
D) 3cos$\left( {{\pi \over 4} - 2\omega t} \right)
100
MediumJEE Mains2021
The time period of a simple pendulum is given by $T = 2\pi \sqrt {{l \over g}} $. The measured value of the length of pendulum is 10 cm known to a 1mm accuracy. The time for 200 oscillations of the pendulum is found to be 100 second using a clock of 1s resolution. The percentage accuracy in the determination of 'g' using this pendulum is 'x'. The value of 'x' to be nearest integer is :-
Options:
A) 2%
B) 3%
C) 5%
D) 4%
101
EasyJEE Mains2021
Two particles A and B of equal masses are suspended from two massless springs of spring constants K1 and K2 respectively. If the maximum velocities during oscillations are equal, the ratio of the amplitude of A and B is
Options:
A) {{{K_1}} \over {{K_2}}}
B) \sqrt {{{{K_1}} \over {{K_2}}}}
C) {{{K_2}} \over {{K_1}}}
D) \sqrt {{{{K_2}} \over {{K_1}}}}
102
EasyJEE Mains2021
A block of mass 1 kg attached to a spring is made to oscillate with an initial amplitude of 12 cm. After 2 minutes the amplitude decreases to 6 cm. Determine the value of the damping constant for this motion . (take ln 2 = 0.693)
Options:
A) 0.69 $\times 102 kg s-$1
B) 3.3 $\times 102 kg s-$1
C) 1.16 $\times 10-2 kg s-$1
D) 5.7 $\times 10-3 kg s-$1
103
EasyJEE Mains2021
For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal ?
Options:
A) x = ${A \over 2}
B) x = $\pm$ A
C) x = $\pm {A \over {\sqrt 2 }}
D) x = 0
104
MediumJEE Mains2021
Amplitude of a mass-spring system, which is executing simple harmonic motion decreases with time. If mass = 500g, Decay constant = 20 g/s then how much time is required for the amplitude of the system to drop to drop to half of its initial value? (ln 2 = 0.693)
Options:
A) 17.32 s
B) 34.65 s
C) 0.034 s
D) 15.01 s
105
EasyJEE Mains2021
Time period of a simple pendulum is T inside a lift when the lift is stationary. If the lift moves upwards with an acceleration g/2, the time period of pendulum will be :
Options:
A) \sqrt 3 T
B) \sqrt {{2 \over 3}} T
C) {T \over {\sqrt 3 }}
D) \sqrt {{3 \over 2}} T
106
EasyJEE Mains2021
A particle executes S.H.M., the graph of velocity as a function of displacement is :
Options:
A) a parabola
B) a helix
C) an ellipse
D) a circle
107
EasyJEE Mains2021
Given below are two statements :Statement I : A second's pendulum has a time period of 1 second.Statement II : It takes precisely one second to move between the two extreme positions.In the light of the above statements, choose the correct answer from the options given below :
Options:
A) Both Statement I and Statement II are false
B) Statement I is false but Statement II is true
C) Both Statement I and Statement II are true
D) Statement I is true but Statement II is false
108
MediumJEE Mains2021
Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance (R/2) from the earth's centre, where 'R' is the radius of the Earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period :
Options:
A) 2\pi \sqrt {{R \over g}}
B) {g \over {2\pi R}}
C) {{2\pi R} \over g}
D) {1 \over {2\pi }}\sqrt {{g \over R}}
109
EasyJEE Mains2021
If two similar springs each of spring constant K1 are joined in series, the new spring constant and time period would be changed by a factor :
Options:
A) {1 \over 2},2\sqrt 2
B) {1 \over 4},2\sqrt 2
C) {1 \over 2},\sqrt 2
D) {1 \over 4},\sqrt 2
110
MediumJEE Mains2021
Y = A sin($\omegat + \phi0) is the time-displacement equation of a SHM. At t = 0 the displacement of the particle is Y = {A \over 2} and it is moving along negative x-direction. Then the initial phase angle \phi$0 will be:
Options:
A) {{5\pi } \over 6}
B) {{\pi } \over 3}
C) {{2\pi } \over 3}
D) {{\pi } \over 6}
111
EasyJEE Mains2021
Two identical springs of spring constant '2k' are attached to a block of mass m and to fixed support (see figure). When the mass is displaced from equilibrium position on either side, it executes simple harmonic motion. The time period of oscillations of this system is :
Options:
A) 2\pi \sqrt {{m \over k}}
B) \pi \sqrt {{m \over k}}
C) 2\pi \sqrt {{m \over {2k}}}
D) \pi \sqrt {{m \over {2k}}}
112
MediumJEE Mains2021
The point A moves with a uniform speed along the circumference of a circle of radius 0.36 m and covers 30$^\circ$ in 0.1 s. The perpendicular projection 'P' from 'A' on the diameter MN represents the simple harmonic motion of 'P'. The restoration force per unit mass when P touches M will be :
Options:
A) 9.87 N
B) 0.49 N
C) 50 N
D) 100 N
113
EasyJEE Mains2021
If the time period of a two meter long simple pendulum is 2s, the acceleration due to gravity at the place where pendulum is executing S.H.M. is :
Options:
A) 16 m/s2
B) 2$\pi2 ms-$2
C) \pi2 ms-$2
D) 9.8 ms$-$2
114
MediumJEE Mains2021
In the given figure, a body of mass M is held between two massless springs, on a smooth inclined plane. The free ends of the springs are attached to firm supports. If each spring has spring constant k, the frequency of oscillation of given body is :
Options:
A) {1 \over {2\pi }}\sqrt {{{2k} \over {Mg\sin \alpha }}}
B) {1 \over {2\pi }}\sqrt {{k \over {Mg\sin \alpha }}}
C) {1 \over {2\pi }}\sqrt {{{2k} \over M}}
D) {1 \over {2\pi }}\sqrt {{k \over {2M}}}
115
EasyJEE Mains2021
When a particle executes SHM, the nature of graphical representation of velocity as a function of displacement is :
Options:
A) circular
B) straight line
C) parabolic
D) elliptical
116
MediumJEE Mains2021
The period of oscillation of a simple pendulum is $T = 2\pi \sqrt {{L \over g}} $. Measured value of 'L' is 1.0 m from meter scale having a minimum division of 1 mm and time of one complete oscillation is 1.95 s measured from stopwatch of 0.01 s resolution. The percentage error in the determination of 'g' will be :
Options:
A) 1.30%
B) 1.33%
C) 1.13%
D) 1.03%
117
MediumJEE Mains2021
In the given figure, a mass M is attached to a horizontal spring which is fixed on one side to a rigid support. The spring constant of the spring is k. The mass oscillates on a frictionless surface with time period T and amplitude A. When the mass is in equilibrium position, as shown in the figure, another mass m is gently fixed upon it. The new amplitude of oscillations will be :
Options:
A) A\sqrt {{M \over {M - m}}}
B) A\sqrt {{{M - m} \over M}}
C) A\sqrt {{{M + m} \over M}}
D) A\sqrt {{M \over {M + m}}}
118
MediumJEE Mains2020
When a particle of mass m is attached to a vertical spring of spring constant k and released, its motion is described by y(t) = y0 sin2 $\omega t, where 'y' is measured from the lower end of unstretched spring. Then \omega $ is:
Options:
A) \sqrt {{g \over {{y_0}}}}
B) {1 \over 2}\sqrt {{g \over {{y_0}}}}
C) \sqrt {{{2g} \over {{y_0}}}}
D) \sqrt {{g \over {2{y_0}}}}
119
MediumJEE Mains2020
A block of mass m attached to a massless spring is performing oscillatory motion of amplitude ‘A’ on a frictionless horizontal plane. If half of the mass of the block breaks off when it is passing through its equilibrium point, the amplitude of oscillation for the remaining system become fA. The value of f is :
Options:
A) 1
B) {1 \over 2}
C) \sqrt 2
D) {1 \over {\sqrt 2 }}
120
MediumJEE Mains2020
The displacement time graph of a particle executing S.H.M is given in figure :(sketch is schematic and not to scale) Which of the following statements is/are true for this motion? (A) The force is zero at t = ${{3T} \over 4} (B) The acceleration is maximum at t = T (C) The speed is maximum at t = {{T} \over 4} (D) The P.E. is equal to K.E. of the oscillation at t = {{T} \over 2}
Options:
A) (B), (C) and (D)
B) (A), (B) and (C)
C) (A) and (D)
D) (A), (B) and (D)
121
MediumJEE Mains2020
A spring mass system (mass m, spring constant k and natural length $l) rest in equilibrium on a horizontal disc. The free end of the spring is fixed at the centre of the disc. If the disc together with spring mass system, rotates about it's axis with an angular velocity \omega , (k \gg m{\omega ^2}$) the relative change in the length of the spring is best given by the option :
Options:
A) {{m{\omega ^2}} \over {3k}}
B) {{m{\omega ^2}} \over k}
C) {{2m{\omega ^2}} \over k}
D) \sqrt {{2 \over 3}} \left( {{{m{\omega ^2}} \over k}} \right)
122
MediumJEE Mains2020
A simple pendulum is being used to determine th value of gravitational acceleration g at a certain place. Th length of the pendulum is 25.0 cm and a stop watch with 1s resolution measures the time taken for 40 oscillations to be 50 s. The accuracy in g is :
Options:
A) 4.40%
B) 3.40%
C) 2.40%
D) 5.40%
123
MediumJEE Mains2019
The displacement of a damped harmonic oscillator is given by x(t ) = e–0.1t cos (10$\pi $t + f). Here t is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to :
Options:
A) 27 s
B) 13 s
C) 7 s
D) 4 s
124
MediumJEE Mains2019
A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to 1/1000 of the original amplitude is close to :-
Options:
A) 100 s
B) 10 s
C) 20 s
D) 50 s
125
MediumJEE Mains2019
A simple harmonic motion is represented by : y = 5 (sin 3 $\pi t + \sqrt 3 cos 3 \pi $t) cm The amplitude and time period of the motion are :
Options:
A) 10 cm, ${3 \over 2}$ s
B) 5 cm, ${2 \over 3}$ s
C) 5 cm, ${3 \over 2}$ s
D) 10 cm, ${2 \over 3}$ s
126
MediumJEE Mains2019
Two light identical springs of spring constant k are attached horizontally at the two ends of a uniform horizontal rod AB of length $\ell $ and mass m. The rod is pivoted at its centre 'O' and can rotate freely in horizontal plane. The other ends of the two springs are fixed to rigid supports as shown in figure. The rod is gently pushed through a small angle and released. The frequency of resulting oscillation is :
Options:
A) {1 \over {2\pi }}\sqrt {{{3k} \over m}}
B) {1 \over {2\pi }}\sqrt {{{6k} \over m}}
C) {1 \over {2\pi }}\sqrt {{k \over m}}
D) {1 \over {2\pi }}\sqrt {{{2k} \over m}}
127
MediumJEE Mains2019
A pendulum is executing simple harmonic motion and its maximum kinetic energy is K1. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is K2. Then :
Options:
A) {K_2} = {{{K_1}} \over 2}
B) K2 = 2K1
C) K2 = K1
D) K2 = ${{{K_1}} \over 4}
128
MediumJEE Mains2019
A simple pendulum of length 1 m is oscillating with an angular frequency 10 rad/s. The support of the pendulum starts oscillating up and down with a small angular frequency of 1 rad/s and an amplitude of 10–2 m. The relative change in the angular frequency of the pendulum is best given by :
Options:
A) 1 rad/s
B) 10$-$3 rad/s
C) 10$-$1 rad/s
D) 10$-$5 rad/s
129
MediumJEE Mains2019
The mass and the diameter of a planet are three times the respective values for the Earth. The period of oscillation of simple pendulum on the Earth is 2 s. The period of oscillation of the same pendulum on the planet would be :
Options:
A) {{\sqrt 3 } \over 2}$ s
B) {3 \over 2}$ s
C) {2 \over {\sqrt 3 }}$ s
D) 2\sqrt 3 $ s
130
MediumJEE Mains2019
A particle undergoing simple harmonic motion has time dependent displacement given by x(t) = Asin${{\pi t} \over {90}}$. The ratio of kinetic to potential energy of this particle at t = 210 s will be:
Options:
A) {1 \over 9}
B) 3
C) 2
D) 1
131
MediumJEE Mains2019
A particle executes simple harmonic motion with an amplitude of 5 cm. When the particle is at 4 cm from the mean position, the magnitude of its velocity in SI units is equal to that of its acceleration. Then, its periodic time in seconds is -
Options:
A) {{4\pi } \over 3}
B) {3 \over 8}\pi
C) {7 \over 3}\pi
D) {{8\pi } \over 3}
132
MediumJEE Mains2019
A closed organ pipe has a fundamental frequency of 1.5 kHz. The number of overtones that can be distinctly heard by a person with this organ pipe will be (Assume that the highest frequency a person can hear is 20,000 Hz)
Options:
A) 4
B) 7
C) 6
D) 5
133
MediumJEE Mains2019
A cylindrical plastic bottle of negligible mass is filled with 310 ml of water and left floating in a pond with still water. If pressed downward slightly and released, it starts performing simple harmonic motion at angular frequency $\omega . If the radius of the bottle is 2.5 cm then \omega $ is close to – (density of water = 103 kg/m3).
Options:
A) 2.50 rad s$-$1
B) 3.75 rad s$-$1
C) 5.00 rad s$-$1
D) 7.90 rad s$-$1
134
MediumJEE Mains2019
A rod of mass 'M' and length '2L' is suspended at its middle by a wire. It exhibits torsional oscillations; If two masses each of 'm' are attached at distance 'L/2' from its centre on both sides, it reduces the oscillation frequency by 20%. The value of radio m/M is close to :
Options:
A) 0.77
B) 0.57
C) 0.37
D) 0.17
135
MediumJEE Mains2019
A particle is executing simple harmonic motion (SHM) of amplitude A, along the x-axis, about x = 0. When its potential Energy (PE) equals kinetic energy (KE), the position of the particle will be :
Options:
A) {A \over 2}
B) {A \over {2\sqrt 2 }}
C) {A \over {\sqrt 2 }}
D) A
136
MediumJEE Mains2018
An oscillator of mass M is at rest in its equilibrium position in a potential V = ${1 \over 2} k(x -$ X)2. A particle of mass m comes from right with speed u and collides completely inelastically with M and sticks to it. This process repeats every time the oscillator crosses its equilibrium position. The amplitude of oscillations after 13 collisions is : (M = 10, m = 5, u = 1, k = 1)
Options:
A) {1 \over {\sqrt 3 }}
B) {1 \over 2}
C) {2 \over 3}
D) {3 \over {\sqrt 5 }}
137
MediumJEE Mains2018
A particle executes simple harmonic motion and is located at x = a, b and c at times t0, 2t0 and 3t0 respectively. The freqquency of the oscillation is :
Options:
A) {1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + c} \over {2b}}} \right)
B) {1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + b} \over {2c}}} \right)
C) {1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{2a + 3c} \over b}} \right)
D) {1 \over {2\,\pi \,{t_0}}}{\cos ^{ - 1}}\left( {{{a + 2b} \over {3c}}} \right)
138
MediumJEE Mains2018
A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of 1012/sec. What is the force constant of the bonds connecting one atom with the other? (Mole wt. of silver = 108 and Avogadro number = 6.02 × 1023 gm mole–1)
Options:
A) 5.5 N/m
B) 6.4 N/m
C) 7.1 N/m
D) 2.2 N/m
139
MediumJEE Mains2018
Two simple harmonic motions, as shown below, are at right angles. They are combined to form Lissajous figures. x(t) = A sin (at + $\delta $) y(t) = B sin (bt) Identify the correct match below.
Options:
A) Parameters A $ \ne B, a = b; \delta $ = 0;
Curve Parabola
B) Parameters A = B, a = b; $\delta = {\raise0.5ex\hbox{\scriptstyle \pi }
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{\scriptstyle 2}}$
Curve Line
C) Parameters A $ \ne B, a = b; \delta = {\raise0.5ex\hbox{\scriptstyle \pi }
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{\scriptstyle 2}}$
Curve Ellipse
D) Parameters A = B, a = 2b; $\delta = {\raise0.5ex\hbox{\scriptstyle \pi }
\kern-0.1em/\kern-0.15em
\lower0.25ex\hbox{\scriptstyle 2}}$
Curve Circle
140
MediumJEE Mains2017
A block of mass 0.1 kg is connected to an elastic spring of spring constant 640 Nm−1 and oscillates in a damping medium of damping constant 10−2 kg s−1 . The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to :
Options:
A) 2 s
B) 3.5 s
C) 5 s
D) 7 s
141
MediumJEE Mains2017
A 1 kg block attached to a spring vibrates with a frequency of 1 Hz on a frictionless horizontal table. Two springs identical to the original spring are attached in parallel to an 8 kg block placed on the same table. So, the frequency of vibration of the 8 kg block is :
Options:
A) {1 \over 4}Hz
B) {1 \over {2\sqrt 2 }}Hz
C) {1 \over 2}Hz
D) 2 Hz
142
MediumJEE Mains2017
The ratio of maximum acceleration to maximum velocity in a simple harmonic motion is 10 s−1 . At, t = 0 the displacement is 5 m. What is the maximum acceleration ? The initial phase is ${\pi \over 4}$.
Options:
A) 500 m/s2
B) 500 $\sqrt 2 m/$ s2
C) 750 m/s2
D) 750 $\sqrt 2 $m / s2
143
MediumJEE Mains2017
A particle is executing simple harmonic motion with a time period T. At time t = 0, it is at its position of equilibrium. The kinetic energy – time graph of the particle will look like:
Options:
A)
B)
C)
D)
144
MediumJEE Mains2016
In an engine the piston undergoes vertical simple harmonic motion with amplitude 7 cm. A washer rests on top of the piston and moves with it. The motor speed is slowly increased. The frequency of the piston at which the washer no longer stays in contact with the piston, is close to :
Options:
A) 0.1 Hz
B) 1.2 Hz
C) 0.7 Hz
D) 1.9 Hz
145
MediumJEE Mains2016
Two particles are performing simple harmonic motion in a straight line about the same equilibrium point. The amplitude and time period for both particles are same and equal to A and I, respectively. At time t = 0 one particle has displacement A while the other one has displacement ${{ - A} \over 2}$ and they are moving towards each other. If they cross each other at time t, then t is :
Options:
A) {T \over 6}
B) {5T \over 6}
C) {T \over 3}
D) {T \over 4}
146
MediumJEE Mains2016
A particle performs simple harmonic motion with amplitude $A. Its speed is trebled at the instant that it is at a distance {{2A} \over 3}$ from equilibrium position. The new amplitude of the motion is:
Options:
A) A\sqrt 3
B) {{7A} \over 3}
C) {A \over 3}\sqrt {41}
D) 3A
147
MediumJEE Mains2015
A pendulum made of a uniform wire of cross sectional area $A has time period T. When an additional mass M is added to its bob, the time period changes to {T_{M.}} If the Young's modulus of the material of the wire is Y then {1 \over Y} is equal to : (g= gravitational acceleration$)
Options:
A) \left[ {1 - {{\left( {{{{T_M}} \over T}} \right)}^2}} \right]{A \over {Mg}}
B) \left[ {1 - {{\left( {{T \over {{T_M}}}} \right)}^2}} \right]{A \over {Mg}}
C) \left[ {{{\left( {{{{T_M}} \over T}} \right)}^2} - 1} \right]{A \over {Mg}}
D) \left[ {{{\left( {{{{T_M}} \over T}} \right)}^2} - 1} \right]{{Mg} \over A}
148
MediumJEE Mains2015
The period of oscillation of a simple pendulum is $T = 2\pi \sqrt {{L \over g}} $. Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using wrist watch of 1 s resolution. The accuracy in the determination of g is:
Options:
A) 1 %
B) 5 %
C) 2 %
D) 3 %
149
MediumJEE Mains2015
For a simple pendulum, a graph is plotted between its kinetic energy $(KE) and potential energy (PE) against its displacement d. Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)
Options:
A)
B)
C)
D)
150
MediumJEE Mains2014
A particle moves with simple harmonic motion in a straight line. In first $\tau s, after starting from rest it travels a distance a, and in next \tau s it travels 2a,$ in same direction, then:
Options:
A) amplitude of motion is $3a
B) time period of oscillations is $8\tau
C) amplitude of motion is $4a
D) time period of oscillations is $6\tau
151
MediumJEE Mains2013
An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M. The piston and the cylinder have equal cross sectional area A. When the piston is in equilibrium, the volume of the gas is {V_0} and its pressure is {P_0}.$ The piston is slightly displaced from the equilibrium position and released,. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frquency
Options:
A) {1 \over {2\pi }}\,{{A\gamma {P_0}} \over {{V_0}M}}
B) {1 \over {2\pi }}\,{{{V_0}M{P_0}} \over {{A^2}\gamma }}
C) {1 \over {2\pi }}\,\sqrt {{{A\gamma {P_0}} \over {{V_0}M}}}
D) {1 \over {2\pi }}\,\sqrt {{{M{V_0}} \over {A\gamma {P_0}}}}
152
MediumJEE Mains2013
The amplitude of a damped oscillator decreases to $0.9 times its original magnitude in 5s. In another 10s it will decrease to \alpha times its original magnitude, where \alpha $ equals
Options:
A) 0.7
B) 0.81
C) 0.729
D) 0.6
153
MediumJEE Mains2012
If a simple pendulum has significant amplitude (up to a factor of $1/e of original ) only in the period between t = 0s\,\,to\,\,t = \tau \,s, then \tau \, may be called the average life of the pendulum When the spherical bob of the pendulum suffers a retardation (due to viscous drag) proportional to its velocity with b$ as the constant of proportionality, the average life time of the pendulum is (assuming damping is small) in seconds :
Options:
A) {{0.693} \over b}
B) b
C) {1 \over b}
D) {2 \over b}
154
MediumJEE Mains2011
Two particles are executing simple harmonic motion of the same amplitude $A and frequency \omega along the x-axis. Their mean position is separated by distance {X_0}\left( {{X_0} > A} \right). If the maximum separation between them is \left( {{X_0} + A} \right),$ the phase difference between their motion is:
Options:
A) {\pi \over 3}
B) {\pi \over 4}
C) {\pi \over 6}
D) {\pi \over 2}
155
MediumJEE Mains2011
A mass $M, attached to a horizontal spring, executes S.H.M. with amplitude {A_1}. When the mass M passes through its mean position then a smaller mass m is placed over it and both of them move together with amplitude {A_2}. The ratio of \left( {{{{A_1}} \over {{A_2}}}} \right)$ is :
Options:
A) {{M + m} \over M}
B) {\left( {{M \over {M + m}}} \right)^{{1 \over 2}}}
C) {\left( {{{M + m} \over M}} \right)^{{1 \over 2}}}
D) {M \over {M + m}}
156
MediumJEE Mains2009
If $x, v and a denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period T,$ then, which of the following does not change with time?
Options:
A) aT/x
B) aT + 2\pi v
C) aT/v
D) {a^2}{T^2} + 4{\pi ^2}{v^2}
157
MediumJEE Mains2007
A particle of mass $m executes simple harmonic motion with amplitude a and frequency v.$ The average kinetic energy during its motion from the position of equilibrium to the end is
Options:
A) 2{\pi ^2}\,m{a^2}{v^2}
B) {\pi ^2}\,m{a^2}{v^2}
C) {1 \over 4}\,m{a^2}{v^2}
D) 4{\pi ^2}m{a^2}{v^2}
158
MediumJEE Mains2007
Two springs, of force constant ${k_1} and {k_2} are connected to a mass m as shown. The frequency of oscillation of the mass is f. If both {k_1} and {k_2}$ are made four times their original values, the frequency of oscillation becomes
Options:
A) 2f
B) f/2
C) f/4
D) 4f
159
MediumJEE Mains2007
The displacement of an object attached to a spring and executing simple harmonic motion is given by $x = 2 \times {10^{ - 2}} cos \pi t$ metre. The time at which the maximum speed first occurs is
Options:
A) 0.25 s
B) 0.5 s
C) 0.75 s
D) 0.125 s
160
MediumJEE Mains2007
A point mass oscillates along the $x-axis according to the law x = {x_0}\,\cos \left( {\omega t - \pi /4} \right). If the acceleration of the particle is written as a = A\,\cos \left( {\omega t + \delta } \right),$ then
Options:
A) A = {x_0}{\omega ^2},\,\,\delta = 3\pi /4
B) A = {x_0},\,\,\delta = - \pi /4
C) A = {x_0}{\omega ^2},\,\,\delta = \pi /4
D) A = {x_0}{\omega ^2},\,\,\delta = - \pi /4
161
MediumJEE Mains2006
Starting from the origin a body oscillates simple harmonically with a period of $2 s. After what time will its kinetic energy be 75\% $ of the total energy?
Options:
A) {1 \over 6}s
B) {1 \over 4}s
C) {1 \over 3}s
D) {1 \over 12}s
162
MediumJEE Mains2006
A coin is placed on a horizontal platform which undergoes vertical simple harmonic motoin of angular frequency $\omega .$ The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time
Options:
A) at the mean position of the platform
B) for an amplitude of ${g \over {{\omega ^2}}}
C) For an amplitude of ${{{g^2}} \over {{\omega ^2}}}
D) at the height position of the platform
163
MediumJEE Mains2006
The maximum velocity of a particle, executing simple harmonic motion with an amplitude $7 mm, is 4.4 m/s.$ The period of oscillation is
Options:
A) 0.01 s
B) 10 s
C) 0.1 s
D) 100 s
164
MediumJEE Mains2005
The function ${\sin ^2}\left( {\omega t} \right)$ represents
Options:
A) a periodic, but not $SHM with a period {\pi \over \omega }
B) a periodic, but not $SHM with a period {{2\pi } \over \omega }
C) a $SHM with a period {\pi \over \omega }
D) a $SHM with a period {{2\pi } \over \omega }
165
MediumJEE Mains2005
Two simple harmonic motions are represented by the equations ${y_1} = 0.1\,\sin \left( {100\pi t + {\pi \over 3}} \right) and {y_2} = 0.1\,\cos \,\pi t. The phase difference of the velocity of particle 1 with respect to the velocity of particle 2$ is
Options:
A) {\pi \over 3}
B) {{ - \pi } \over 6}
C) {\pi \over 6}
D) {{ - \pi } \over 3}
166
MediumJEE Mains2005
If a simple harmonic motion is represented by ${{{d^2}x} \over {d{t^2}}} + \alpha x = 0.$ its time period is
Options:
A) {{2\pi } \over {\sqrt \alpha }}
B) {{2\pi } \over \alpha }
C) 2\pi \sqrt \alpha
D) 2\pi \alpha
167
MediumJEE Mains2005
The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would
Options:
A) first decrease and then increase to the original value
B) first increase and then decrease to the original value
C) increase towards a saturation value
D) remain unchanged
168
MediumJEE Mains2004
In forced oscillation of a particle the amplitude is maximum for a frequency ${\omega _1} of the force while the energy is maximum for a frequency {\omega _2}$ of the force; then
Options:
A) {\omega _1} < {\omega _2} when damping is small and {\omega _1} > {\omega _2}$ when damping is large
B) {\omega _1} > {\omega _2}
C) {\omega _1} = {\omega _2}
D) {\omega _1} < {\omega _2}
169
MediumJEE Mains2004
A particle of mass $m is attached to a spring (of spring constant k) and has a natural angular frequency {\omega _0}. An external force F(t) proportional to \cos \,\omega t\left( {\omega \ne {\omega _0}} \right)$ is applied to the oscillator. The time displacement of the oscillator will be proportional to
Options:
A) {1 \over {m\left( {\omega _0^2 + {\omega ^2}} \right)}}
B) {1 \over {m\left( {\omega _0^2 - {\omega ^2}} \right)}}
C) {m \over {\omega _0^2 - {\omega ^2}}}
D) {m \over {\omega _0^2 + {\omega ^2}}}
170
MediumJEE Mains2004
The total energy of particle, executing simple harmonic motion is
Options:
A) independent of $x
B) \propto \,{x^2}
C) \propto \,x
D) \propto \,{x^{1/2}}
171
MediumJEE Mains2004
A particle at the end of a spring executes $S.H.M with a period {t_1}. While the corresponding period for another spring is {t_2}. If the period of oscillation with the two springs in series is T$ then
Options:
A) {T^{ - 1}} = t_1^{ - 1} + t_2^{ - 1}
B) {T^2} = t_1^2 + t_2^2
C) T = {t_1} + {t_2}
D) {T^{ - 2}} = t_1^{ - 2} + t_2^{ - 2}
172
MediumJEE Mains2004
The bob of a simple pendulum executes simple harmonic motion in water with a period $t, while the period of oscillation of the bob is {t_0} in air. Neglecting frictional force of water and given that the density of the bob is \left( {4/3} \right) \times 1000\,\,kg/{m^3}. What relationship between t and {t_0}$ is true
Options:
A) t = 2{t_0}
B) t = {t_0}/2
C) t = {t_0}
D) t = 4{t_0}
173
MediumJEE Mains2003
Two particles $A and B of equal masses are suspended from two massless springs of spring of spring constant {k_1} and {k_2}, respectively. If the maximum velocities, during oscillation, are equal, the ratio of amplitude of A and B$ is
Options:
A) \sqrt {{{{k_1}} \over {{k_2}}}}
B) {{{{k_2}} \over {{k_1}}}}
C) \sqrt {{{{k_2}} \over {{k_1}}}}
D) {{{{k_1}} \over {{k_2}}}}
174
MediumJEE Mains2003
The length of a simple pendulum executing simple harmonic motion is increased by $21\% $. The percentage increase in the time period of the pendulum of increased length is
Options:
A) 11\%
B) 21\%
C) 42\%
D) 10\%
175
MediumJEE Mains2003
A mass $M is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes SHM of time period T. If the mass is increased by m. the time period becomes {{5T} \over 3}. Then the ratio of {{m} \over M}$ is
Options:
A) {3 \over 5}
B) {25 \over 9}
C) {16 \over 9}
D) {5 \over 3}
176
MediumJEE Mains2003
A body executes simple harmonic motion. The potential energy $(P.E), the kinetic energy (K.E) and total energy (T.E) are measured as a function of displacement x.$ Which of the following statements is true ?
Options:
A) K.E is maximum when x=0
B) T.E is zero when x=0
C) K.E is maximum when x$ is maximum
D) P.E is maximum when x=0
177
MediumJEE Mains2003
The displacement of particle varies according to the relation $x=4\left( {\cos \,\pi t + \sin \,\pi t} \right).$ The amplitude of the particle is
Options:
A) -4
B) 4
C) 4\sqrt 2
D) 8
178
MediumJEE Mains2002
In a simple harmonic oscillator, at the mean position
Options:
A) kinetic energy is minimum, potential energy is maximum
B) both kinetic and potential energies are maximum
C) kinetic energy is maximum, potential energy is minimum
D) both kinetic and potential energies are minimum.
179
MediumJEE Mains2002
A child swinging on a swing in sitting position, stands up, then the time period of the swing will
Options:
A) increase
B) decrease
C) remains same
D) increases of the child is long and decreases if the child is short
180
MediumJEE Mains2002
If a spring has time period $T, and is cut into n$ equal parts, then the time period of each part will be
Options:
A) T\sqrt n
B) T/\sqrt n
C) nT
D) T
181
MediumJEE Mains2026
The displacement of a particle, executing simple harmonic motion with time period T, is expressed as x(t)=A \sin \omega t, where A is the amplitude. The maximum value of potential energy of this oscillator is found at t=T / 2 \beta. The value of \beta is \_\_\_\_ .
Options:
182
EasyJEE Mains2024
A particle of mass $0.50 \mathrm{~kg} executes simple harmonic motion under force F=-50(\mathrm{Nm}^{-1}) x. The time period of oscillation is \frac{x}{35} s. The value of x is _________. (Given \pi=\frac{22}{7}$)
Options:
183
MediumJEE Mains2024
The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes of $4 \mathrm{~m}, 2 \mathrm{~ms}^{-1} and 16 \mathrm{~ms}^{-2} at a certain instant. The amplitude of the motion is \sqrt{x}, \mathrm{~m} where x$ is _________.
Options:
184
MediumJEE Mains2024
An object of mass $0.2 \mathrm{~kg} executes simple harmonic motion along x axis with frequency of \left(\frac{25}{\pi}\right) \mathrm{Hz}. At the position x=0.04 \mathrm{~m} the object has kinetic energy 0.5 \mathrm{~J} and potential energy 0.4 \mathrm{~J}. The amplitude of oscillation is ________ \mathrm{cm}$.
Options:
185
EasyJEE Mains2024
A particle is doing simple harmonic motion of amplitude $0.06 \mathrm{~m} and time period 3.14 \mathrm{~s}. The maximum velocity of the particle is _________ \mathrm{cm} / \mathrm{s}$.
Options:
186
EasyJEE Mains2024
The displacement of a particle executing SHM is given by $x=10 \sin \left(w t+\frac{\pi}{3}\right) m. The time period of motion is 3.14 \mathrm{~s}. The velocity of the particle at t=0 is _______ \mathrm{m} / \mathrm{s}$.
Options:
187
EasyJEE Mains2024
A mass m is suspended from a spring of negligible mass and the system oscillates with a frequency f_1. The frequency of oscillations if a mass 9 \mathrm{~m} is suspended from the same spring is f_2. The value of \frac{f_1}{f_2} \mathrm{i} ________.
Options:
188
MediumJEE Mains2024
The time period of simple harmonic motion of mass $M in the given figure is \pi \sqrt{\frac{\alpha M}{5 k}}, where the value of \alpha$ is _________.
Options:
189
MediumJEE Mains2024
A particle performs simple harmonic motion with amplitude $A. Its speed is increased to three times at an instant when its displacement is \frac{2 A}{3}. The new amplitude of motion is \frac{n A}{3}. The value of n$ is ___________.
Options:
190
MediumJEE Mains2024
A simple harmonic oscillator has an amplitude $A and time period 6 \pi second. Assuming the oscillation starts from its mean position, the time required by it to travel from x= A to x=\frac{\sqrt{3}}{2} A will be \frac{\pi}{x} \mathrm{~s}, where x=$ _________.
Options:
191
EasyJEE Mains2024
When the displacement of a simple harmonic oscillator is one third of its amplitude, the ratio of total energy to the kinetic energy is $\frac{x}{8}, where x=$ _________.
Options:
192
EasyJEE Mains2024
A particle executes simple harmonic motion with an amplitude of $4 \mathrm{~cm}. At the mean position, velocity of the particle is 10 \mathrm{~cm} / \mathrm{s}. The distance of the particle from the mean position when its speed becomes 5 \mathrm{~cm} / \mathrm{s} is \sqrt{\alpha} \mathrm{~cm}, where \alpha=$ ________.
Options:
193
MediumJEE Mains2023
At a given point of time the value of displacement of a simple harmonic oscillator is given as $\mathrm{y}=\mathrm{A} \cos \left(30^{\circ}\right). If amplitude is 40 \mathrm{~cm} and kinetic energy at that time is 200 \mathrm{~J}, the value of force constant is 1.0 \times 10^{x} ~\mathrm{Nm}^{-1}. The value of x$ is ____________.
Options:
194
EasyJEE Mains2023
A rectangular block of mass $5 \mathrm{~kg} attached to a horizontal spiral spring executes simple harmonic motion of amplitude 1 \mathrm{~m} and time period 3.14 \mathrm{~s}$. The maximum force exerted by spring on block is _________ N
Options:
195
MediumJEE Mains2023
A simple pendulum with length $100 \mathrm{~cm} and bob of mass 250 \mathrm{~g} is executing S.H.M. of amplitude 10 \mathrm{~cm}. The maximum tension in the string is found to be \frac{x}{40} \mathrm{~N}. The value of x$ is ___________.
Options:
196
EasyJEE Mains2023
The amplitude of a particle executing SHM is $3 \mathrm{~cm}. The displacement at which its kinetic energy will be 25 \% more than the potential energy is: __________ \mathrm{cm}
Options:
197
EasyJEE Mains2023
In the figure given below, a block of mass $M=490 \mathrm{~g} placed on a frictionless table is connected with two springs having same spring constant \left(\mathrm{K}=2 \mathrm{~N} \mathrm{~m}^{-1}\right). If the block is horizontally displaced through '\mathrm{X}' \mathrm{m} then the number of complete oscillations it will make in 14 \pi$ seconds will be _____________.
Options:
198
MediumJEE Mains2023
The velocity of a particle executing SHM varies with displacement (x) as 4 v^{2}=50-x^{2}. The time period of oscillations is \frac{x}{7} s. The value of x is ___________. \left(\right. Take \left.\pi=\frac{22}{7}\right)
Options:
199
MediumJEE Mains2023
The general displacement of a simple harmonic oscillator is $x = A\sin \omega t. Let T be its time period. The slope of its potential energy (U) - time (t) curve will be maximum when t = {T \over \beta }. The value of \beta$ is ______________.
Options:
200
EasyJEE Mains2023
A particle of mass 250 g executes a simple harmonic motion under a periodic force $\mathrm{F}=(-25~x)\mathrm{N}$. The particle attains a maximum speed of 4 m/s during its oscillation. The amplitude of the motion is ___________ cm.
Options:
201
EasyJEE Mains2023
A mass m attached to free end of a spring executes SHM with a period of 1s. If the mass is increased by 3 kg the period of the oscillation increases by one second, the value of mass m is ___________ kg.
Options:
202
EasyJEE Mains2023
A block of a mass 2 kg is attached with two identical springs of spring constant 20 N/m each. The block is placed on a frictionless surface and the ends of the springs are attached to rigid supports (see figure). When the mass is displaced from its equilibrium position, it executes a simple harmonic motion. The time period of oscillation is $\frac{\pi}{\sqrt x} in SI unit. The value of x$ is ____________.
Options:
203
MediumJEE Mains2022
The metallic bob of simple pendulum has the relative density 5. The time period of this pendulum is $10 \mathrm{~s}. If the metallic bob is immersed in water, then the new time period becomes 5 \sqrt{x} s. The value of x$ will be ________.
Options:
204
MediumJEE Mains2022
The potential energy of a particle of mass $4 \mathrm{~kg} in motion along the x-axis is given by \mathrm{U}=4(1-\cos 4 x) J. The time period of the particle for small oscillation (\sin \theta \simeq \theta) is \left(\frac{\pi}{K}\right) s. The value of \mathrm{K}$ is _________.
Options:
205
MediumJEE Mains2022
A mass $0.9 \mathrm{~kg}, attached to a horizontal spring, executes SHM with an amplitude \mathrm{A}_{1}. When this mass passes through its mean position, then a smaller mass of 124 \mathrm{~g} is placed over it and both masses move together with amplitude A_{2}. If the ratio \frac{A_{1}}{A_{2}} is \frac{\alpha}{\alpha-1}, then the value of \alpha$ will be ___________.
Options:
206
MediumJEE Mains2022
As per given figures, two springs of spring constants $k and 2 k are connected to mass m. If the period of oscillation in figure (a) is 3 \mathrm{s}, then the period of oscillation in figure (b) will be \sqrt{x}~ s. The value of x$ is ___________.
Options:
207
MediumJEE Mains2022
A body is performing simple harmonic with an amplitude of 10 cm. The velocity of the body was tripled by air jet when it is at 5 cm from its mean position. The new amplitude of vibration is $\sqrt{x}$ cm. The value of x is _____________.
Options:
208
EasyJEE Mains2022
A pendulum is suspended by a string of length 250 cm. The mass of the bob of the pendulum is 200 g. The bob is pulled aside until the string is at 60$^\circ with vertical as shown in the figure. After releasing the bob, the maximum velocity attained by the bob will be ____________ ms-$1. (if g = 10 m/s2)
Options:
209
EasyJEE Mains2022
A particle executes simple harmonic motion. Its amplitude is 8 cm and time period is 6 s. The time it will take to travel from its position of maximum displacement to the point corresponding to half of its amplitude, is ___________ s.
Options:
210
MediumJEE Mains2021
A particle of mass 1 kg is hanging from a spring of force constant 100 Nm$-1. The mass is pulled slightly downward and released so that it executes free simple harmonic motion with time period T. The time when the kinetic energy and potential energy of the system will become equal, is {T \over x}$. The value of x is _____________.
Options:
211
EasyJEE Mains2021
Two simple harmonic motion, are represented by the equations ${y_1} = 10\sin \left( {3\pi t + {\pi \over 3}} \right) {y_2} = 5(\sin 3\pi t + \sqrt 3 \cos 3\pi t)$ Ratio of amplitude of y1 to y2 = x : 1. The value of x is ______________.
Options:
212
EasyJEE Mains2021
Two simple harmonic motions are represented by the equations ${x_1} = 5\sin \left( {2\pi t + {\pi \over 4}} \right) and {x_2} = 5\sqrt 2 (\sin 2\pi t + \cos 2\pi t)$. The amplitude of second motion is ................ times the amplitude in first motion.
Options:
213
EasyJEE Mains2021
A particle executes simple harmonic motion represented by displacement function as x(t) = A sin($\omegat + \phi)If the position and velocity of the particle at t = 0 s are 2 cm and 2\omega cm s-1 respectively, then its amplitude is x\sqrt 2 $ cm where the value of x is _________________.
Options:
214
MediumJEE Mains2021
In the reported figure, two bodies A and B of masses 200 g and 800 g are attached with the system of springs. Springs are kept in a stretched position with some extension when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be ____________ rad/s when k = 20 N/m.
Options:
215
EasyJEE Mains2021
A pendulum bob has a speed of 3 m/s at its lowest position. The pendulum is 50 cm long. The speed of bob, when the length makes an angle of 60$^\circ$ to the vertical will be (g = 10 m/s2) ____________ m/s.
Options:
216
EasyJEE Mains2021
A particle performs simple harmonic motion with a period of 2 second. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is ${1 \over a}$s. The value of 'a' to the nearest integer is _________.
Options:
217
EasyJEE Mains2021
Consider two identical springs each of spring constant k and negligible mass compared to the mass M as shown. Fig. 1 shows one of them and Fig. 2 shows their series combination. The ratios of time period of oscillation of the two SHM is Tb/Ta = $\sqrt x $, where value of x is ___________. (Round off to the Nearest Integer)
Options:
218
MediumJEE Mains2021
A particle executes S.H.M. with amplitude 'a', and time period 'T'. The displacement of the particle when its speed is half of maximum speed is ${{\sqrt x a} \over 2}$. The value of x is __________.
Options:
219
EasyJEE Mains2021
Time period of a simple pendulum is T. The time taken to complete ${5 \over 8} oscillations starting from mean position is {\alpha \over \beta }T. The value of \alpha$ is _________.
Options:
220
MediumMHT CET2025
All the springs in fig. (a), (b) and (c) are identical, each having force constant K each. Mass m is attached to each system. If \mathrm{T}_a, \mathrm{~T}_b and \mathrm{T}_{\mathrm{c}} are the time periods of oscillations of the three systems in fig. (a), (b) and (c) respectively, then
Options:
A) \quad \mathrm{T}_{\mathrm{a}}=\sqrt{2} \mathrm{~T}_{\mathrm{b}}
B) \quad \mathrm{T}_{\mathrm{a}}=\frac{\mathrm{T}_{\mathrm{c}}}{\sqrt{2}}
C) \mathrm{T}_{\mathrm{b}}=2 \mathrm{~T}_{\mathrm{a}}
D) \quad \mathrm{T}_{\mathrm{b}}=2 \mathrm{~T}_{\mathrm{c}}
221
MediumMHT CET2025
A point particle of mass 200 gram is executing S.H.M. of amplitude 0.2 m . When the particle passes through the mean position, its kinetic energy is 16 \times 10^{-3} \mathrm{~J}. The equation of motion of this particle is (Initial phase of oscillation =0^{\circ} )
Options:
A) \mathrm{Y}=0.2 \sin (4 \mathrm{t})
B) \mathrm{Y}=0.2 \sin \left(\frac{\mathrm{t}}{4}\right)
C) \mathrm{Y}=0.2 \sin \left(\frac{\mathrm{t}}{2}\right)
D) \mathrm{Y}=0.2 \sin (2 \mathrm{t})
222
MediumMHT CET2025
A simple pendulum starts oscillating simple harmonically from its mean position ( \mathrm{x}=0 ) with amplitude ' a ' and periodic time ' T '. The magnitude of velocity of pendulum at x=\frac{a}{2} is
Options:
A) \frac{3 \pi^2 a}{T}
B) \frac{\sqrt{3} \pi a}{2 T}
C) \frac{\pi a}{T}
D) \frac{\sqrt{3} \pi \mathrm{a}}{\mathrm{T}}
223
MediumMHT CET2025
A simple pendulum is suspended from ceiling of a lift when lift is at rest its period is ' T '. With what acceleration ' a ' should lift be accelerated upward in order to reduce the period to ' T '? (take ' g ' as acceleration due to gravity)
Options:
A) 2 g
B) 3 g
C) 4 g
D) g
224
MediumMHT CET2025
A particle is performing S.H.M. starting from extreme position. Graphical representation shows that between displacement and acceleration, there is a phase difference of
Options:
A) \pi \mathrm{rad}
B) \frac{\pi}{2} \mathrm{rad}
C) \frac{\pi}{4} \mathrm{rad}
D) 0 rad
225
MediumMHT CET2025
A mass ' M ' attached to a horizontal spring executes S.H.M. of amplitude A_1. When the mass M passes through its mean position, then a smaller mass ' m ' is placed over it and both of them move together with amplitude \mathrm{A}_2. The ratio \left(\frac{A_1}{A_2}\right) is
Options:
A) \frac{M+m}{M}
B) \frac{\mathrm{M}}{\mathrm{M}+\mathrm{m}}
C) \left(\frac{M+m}{M}\right)^{\frac{1}{2}}
D) \left(\frac{M}{M+m}\right)^{\frac{1}{2}}
226
MediumMHT CET2025
At a place, the length of the oscillating simple pendulum is made \frac{1}{4} times keeping amplitude same then the total energy will be
Options:
A) 2 times
B) 4 times
C) 8 times
D) 16 times
227
MediumMHT CET2025
A spring executes S.H.M. with mass 1 kg attached to it. The force constant of the spring is 4 \mathrm{~N} / \mathrm{m}. If at any instant its velocity is 20 \mathrm{~cm} / \mathrm{s}, the displacement at that instant is (Amplitude of S.H.M. is 0.4 m )
Options:
A) \sqrt{0.11} \mathrm{~m}
B) \sqrt{0.15} \mathrm{~m}
C) \sqrt{0.17} \mathrm{~m}
D) \sqrt{0.19} \mathrm{~m}
228
MediumMHT CET2025
The ratio of the frequencies of two simple pendulums is 4: 3 at the same place. The ratio of their respective lengths is
Options:
A) 3: 4
B) 4: 3
C) 9: 16
D) 16: 9
229
MediumMHT CET2025
Two simple pendulums have first (A) bob of mass ' M_1 ' and length ' L_1 ', second (B) of mass ' \mathrm{M}_2 ' and length ' \mathrm{L}_2 '. \mathrm{M}_1=\mathrm{M}_2 and \mathrm{L}_1=2 \mathrm{~L}_2. If their total energies are same then the correct statement is
Options:
A) amplitude of B is greater than amplitude of A.
B) amplitude of B is smaller than amplitude of A.
C) amplitude of both will be same.
D) amplitude of B is twice that of A .
230
MediumMHT CET2025
As shown in the figure, S_1 and S_2 are identical springs with spring constant K each. The oscillation frequency of the mass ' m ' is ' f '. If the spring \mathrm{S}_2 is removed, the oscillation frequency will become
Options:
A) f
B) 2 f
C) \frac{\mathrm{f}}{\sqrt{2}}
D) \sqrt{2} \cdot \mathrm{f}
231
MediumMHT CET2025
A particle starts oscillating simple harmonically from its mean position with time period ' T '. At time \mathrm{t}=\frac{\mathrm{T}}{6}, the ratio of the potential energy to kinetic energy of the particle is $ \left[\sin 30^{\circ}=\cos 60^{\circ}=0 \cdot 5, \cos 30^{\circ}=\sin 60^{\circ}=\sqrt{3} / 2\right]
Options:
A) 1: 2
B) 1: 3
C) 2: 1
D) 3: 1
232
MediumMHT CET2025
Two particles ' A ' and ' B ' execute SHMs of periods ' T ' and \frac{3 T}{2}. If they start from the mean position then the phase difference between them, when the particle ' A ' completes two oscillations will be
Options:
A) \frac{\pi}{3}
B) \frac{\pi}{6}
C) \frac{4 \pi}{3}
D) \frac{\pi}{4}
233
MediumMHT CET2025
A small sphere oscillates simple harmonically in a watch glass whose radius of curvature is 1.6 m . The period of oscillation of the sphere is (acceleration due to gravity \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 )
Options:
A) 0.2 \pi \mathrm{~s}
B) 0.4 \pi \mathrm{~s}
C) 0.6 \pi \mathrm{~s}
D) 0.8 \pi \mathrm{~s}
234
MediumMHT CET2025
The displacement of particle in S.H.M. is \mathrm{x}=\mathrm{A} \cos (\omega \mathrm{t}+\pi / 6). Its speed will be maximum at time \left(\sin 90^{\circ}=1\right)
Options:
A) \frac{\pi}{3 \omega} \mathrm{~s}
B) \frac{\pi}{2 \omega} \mathrm{~s}
C) \frac{\pi}{\omega} \mathrm{s}
D) \frac{\pi}{4 \omega} \mathrm{~s}
235
MediumMHT CET2025
An object of mass 0.2 kg executes simple harmonic oscillations along the x -axis with frequency of \left(\frac{25}{\pi}\right) \mathrm{Hz}. At the position x=0.04 \mathrm{~m}, the object has kinetic energy 1 J and potential energy 0.6 J . The amplitude of oscillation is
Options:
A) 0.06 m
B) 0.6 m
C) 0.08 m
D) 0.8 m
236
MediumMHT CET2025
The motion of the particle is given by the equation \mathrm{x}=\mathrm{A} \sin \omega \mathrm{t}+\mathrm{B} \cos \omega \mathrm{t}. The motion of the particle is
Options:
A) simple harmonic with amplitude (\mathrm{A}+\mathrm{B})
B) simple harmonic with amplitude (A-B)
C) simple harmonic with amplitude \left(A^2+B^2\right)^{\frac{1}{2}}
D) not simple harmonic
237
MediumMHT CET2025
A particle is executing S.H.M. of amplitude ' A '. When the potential energy of the particle is half of its maximum value during the oscillation, its displacement from the equilibrium position is
Options:
A) \pm \frac{\mathrm{A}}{4}
B) \pm \frac{A}{2}
C) \pm \frac{\mathrm{A}}{\sqrt{3}}
D) \pm \frac{A}{\sqrt{2}}
238
MediumMHT CET2025
A particle is executing linear S.H.M. starting from mean position. The ratio of the kinetic energy to the potential energy of the particle at a point of half the amplitude is
Options:
A) 2: 1
B) 3: 1
C) 4: 1
D) 8: 1
239
MediumMHT CET2025
The amplitude of a damped oscillator becomes \left(\frac{1}{3}\right)^{\mathrm{rd}} of original amplitude in 2 seconds. If its amplitude after 6 second become \left(\frac{1}{n}\right) times the original amplitude, the value of n is ( n is non zero integer)
Options:
A) 9
B) 3
C) 81
D) 27
240
MediumMHT CET2025
A small spherical ball of radius ' r ' is rolling on a curved surface which is frictionless and has a radius of curvature ' R '. Its motion is simple harmonic. Then its tine period of oscillation is proportional to ( \mathrm{g}= acceleration due to gravity)
Options:
A) \sqrt{\frac{\mathrm{R}}{\mathrm{g}}}
B) \sqrt{\frac{\mathrm{r}}{\mathrm{g}}}
C) \sqrt{\frac{R-r}{g}}
D) \sqrt{\frac{R+r}{g}}
241
MediumMHT CET2025
A particle executes S.H.M. starting from the mean position. Its amplitude is ' a ' and its periodic time is ' T '. At a certain instant, its speed ' u ' is half that of maximum speed \mathrm{V}_{\text {max }}. The displacement of the particle at that instant is
Options:
A) \frac{2 \mathrm{a}}{\sqrt{3}}
B) \frac{\sqrt{2} \mathrm{a}}{3}
C) \frac{3 \mathrm{a}}{\sqrt{2}}
D) \frac{\sqrt{3} \mathrm{a}}{2}
242
MediumMHT CET2025
A particle performing linear S.H.M. has period 8 seconds. At time \mathrm{t}=0, it is in the mean position. The ratio of the distances travelled by the particle in the 1^{\text {st }} and 2^{\text {nd }} second is \left(\cos 45^{\circ}=1 / \sqrt{2}\right)
Options:
A) 1:(\sqrt{2}-1)
B) 1: 2
C) 2: 1
D) 1:(\sqrt{2}+1)
243
MediumMHT CET2025
For a particle performing S.H.M.; the total energy is ' n ' times the kinetic energy, when the displacement of a particle from mean position is \frac{\sqrt{3}}{2} \mathrm{~A}, where A is the amplitude of S.H.M. The value of ' n ' is
Options:
A) 2
B) 3
C) 4
D) 6
244
MediumMHT CET2025
The length of the simple pendulum is made 3 times the original length. If ' T ' is its original time period, then the new time period will be
Options:
A) 3 T
B) \quad \sqrt{3} \mathrm{~T}
C) \frac{\mathrm{T}}{\sqrt{3}}
D) \frac{\mathrm{T}}{3}
245
MediumMHT CET2025
Two simple harmonic motions of angular frequency 300 \mathrm{rad} / \mathrm{s} and 3000 \mathrm{rad} / \mathrm{s} have same amplitude. The ratio of their maximum accelerations is
Options:
A) 1: 10
B) 1: 10^2
C) 1: 10^3
D) 1: 10^4
246
MediumMHT CET2025
A mass m is suspended from a spring of negligible mass. The spring is pulled a little and then released, so that mass executes S.H.M. of time period T. If the mass is increased by m_0, the periodic time becomes \frac{5 \mathrm{~T}}{4}. The ratio \frac{\mathrm{m}_0}{\mathrm{M}}
Options:
A) \frac{3}{4}
B) \frac{4}{3}
C) \frac{9}{16}
D) \frac{16}{9}
247
MediumMHT CET2025
A simple pendulum has time period ' \mathrm{T}_1 '. The point of suspension is now moved upward according to equation \mathrm{y}=\mathrm{kt}^2 where \mathrm{k}=1 \mathrm{~m} / \mathrm{s}^2. If new time period is ' \mathrm{T}_2 ' then \mathrm{T}_1^2 / \mathrm{T}_2^2 will be ( \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 )
Options:
A) 2 / 3
B) 5 / 6
C) 6 / 5
D) 3 / 2
248
MediumMHT CET2025
A mass attached to a spring performs S.H.M. whose displacement is \mathrm{x}=3 \times 10^{-3} \cos 2 \pi \mathrm{t} metre. The time taken to obtain maximum speed for the first time is
Options:
A) \frac{1}{12} \mathrm{~s}
B) \frac{1}{8} \mathrm{~s}
C) \frac{1}{4} \mathrm{~s}
D) \frac{1}{2} \mathrm{~s}
249
MediumMHT CET2025
A particle executes a linear S.H.M. In two of its positions the velocities are \mathrm{V}_1, \mathrm{~V}_2 and accelerations are \mathrm{a}_1 and \mathrm{a}_2 respectively $\left(0
Options:
A) \frac{V_1^2-V_2^2}{a_1-a_2}
B) \frac{V_2^2-V_1^2}{a_1-a_2}
C) \frac{V_1^2-V_2^2}{a_1+a_2}
D) \frac{v_2^2-v_1^2}{\left(a_1^2+a_2^2\right)}
250
MediumMHT CET2025
A mass x gram is suspended from a light spring. It is pulled in downward direction and released so that mass performs S.H.M. of period T. If mass is increased by Y gram, the period becomes \frac{4 \mathrm{~T}}{3}. The ratio of \mathrm{Y} / \mathrm{x} is
Options:
A) 7: 9
B) 5: 4
C) 3: 2
D) 8: 7
251
MediumMHT CET2025
The period of S. H.M. of a particle is 16 second. The phase difference between the positions at \mathrm{t}=2 \mathrm{~s} and \mathrm{t}=4 \mathrm{~s} will be
Options:
A) \pi
B) \frac{\pi}{2}
C) \frac{\pi}{4}
D) \frac{\pi}{8}
252
MediumMHT CET2025
If the period of a oscillation of mass ' m ' suspended from a spring is 2 s , then the period of suspended mass ' 4 m ' with the same spring will be
Options:
A) 1 s
B) 3 s
C) 2 s
D) 4 s
253
MediumMHT CET2025
A particle oscillates in straight line simple harmonically with period 8 second and amplitude 4 \sqrt{2} \mathrm{~m}. Particle starts from mean position. The ratio of the distance travelled by it in 1^{\text {st }} second of its motion to that in 2^{\text {nd }} second is \left(\sin 45^{\circ}=1 / \sqrt{2}, \sin \frac{\pi}{2}=1\right)
Options:
A) 1: 8
B) 1: 4
C) 1: 2
D) 1:(\sqrt{2}-1)
254
MediumMHT CET2025
A vertical spring oscillates with period 6 second with mass m is suspended from it. When the mass is at rest, the spring is stretched through a distance of (Take, acceleration due to gravity, \mathrm{g}=\pi^2=10 \mathrm{~m} / \mathrm{s}^2 )
Options:
A) 10 m
B) 3 m
C) 6 m
D) 9 m
255
MediumMHT CET2025
For a particle performing; S.H.M. the displacement - time graph is shown. For that particle the force - time graph is correctly shown in graph
Options:
A) (a)
B) (b)
C) (c)
D) (d)
256
MediumMHT CET2025
If the length of the oscillating simple pendulum is made \frac{1}{3} times the original keeping amplitude same then increase in its total energy at a place will be
Options:
A) 3 times
B) 2 times
C) 9 times
D) 5 times
257
MediumMHT CET2025
The time period of a simple pendulum inside a stationary lift is \sqrt{3} second. When the lift moves upwards with an acceleration g / 3, the time period will be ( \mathrm{g}= acceleration due to gravity)
Options:
A) 1.5 s
B) 2 s
C) \sqrt{3} \mathrm{~s}
D) 3 s
258
MediumMHT CET2025
A mass suspended from a vertical spring performs S.H.M. of period 0.1 second. The spring is unstretched at the highest point of suspension. Maximum speed of the mass is (Gravitational acceleration \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 )
Options:
A) \frac{1}{2 \pi} \mathrm{~m} / \mathrm{s}
B) \frac{1}{\pi} \mathrm{~m} / \mathrm{s}
C) \frac{2}{\pi} \mathrm{~m} / \mathrm{s}
D) \pi \mathrm{m} / \mathrm{s}
259
MediumMHT CET2025
' P ' and ' Q ' are fixed points in same plane and mass ' m ' is tied by string as shown in figure. If the mass is displaced slightly out of this plane and released, it will oscillate with time period (\mathrm{PQ}=2 \mathrm{~d}, \mathrm{PR}=\mathrm{QR}=\mathrm{L})(\mathrm{g}= gravitational acceleration)
Options:
A) 2 \pi \sqrt{\frac{\mathrm{~L}}{\mathrm{~g}}}
B) 2 \pi \sqrt{\frac{\mathrm{~L}^2}{\mathrm{~g}}}
C) 2 \pi \sqrt{\frac{\left(L^2-d^2\right)^{1 / 2}}{g}}
D) 2 \pi \sqrt{\frac{\left(L^2+d^2\right)^{1 / 2}}{g}}
260
MediumMHT CET2024
The bob of a pendulum of length ' l ' is pulled aside from its equilibrium position through an angle ' \theta ' and then released. The bob will then pass through its equilibrium position with speed ' v ', where ' v ' equal to ( g= acceleration due to gravity)
Options:
A) \sqrt{2 g l(1-\cos \theta)}
B) \sqrt{2 \mathrm{~g} l(1+\sin \theta)}
C) \sqrt{2 g l(1-\sin \theta)}
D) \sqrt{2 \mathrm{~g} l(1+\cos \theta)}
261
MediumMHT CET2024
The kinetic energy of a particle, executing simple harmonic motion is 16 J when it is in mean position. If amplitude of motion is 25 cm and the mass of the particle is 5.12 kg , the period of oscillation is
Options:
A) \frac{\pi}{5} \mathrm{~s}
B) 2 \pi \mathrm{~s}
C) 20 \pi \mathrm{~s}
D) 5 \pi \mathrm{~s}
262
MediumMHT CET2024
A particle performs linear S.H.M. At a particular instant, velocity of the particle is ' u ' and acceleration is ' \alpha ' while at another instant, velocity is ' v ' and acceleration is ' \beta ' (0<\alpha<\beta). The distance between the two positions is
Options:
A) \frac{u^2-v^2}{\alpha+\beta}
B) \frac{u^2+v^2}{\alpha+\beta}
C) \frac{u^2-v^2}{\alpha-\beta}
D) \frac{u^2+v^2}{\alpha-\beta}
263
MediumMHT CET2024
A particle executing S.H.M. has velocities ' \mathrm{V}_1 ' and ' \mathrm{V}_2 ' at distances ' x_1 ' and ' x_2 ' respectively, from the mean position. Its frequency is
Options:
A) \frac{1}{2 \pi} \sqrt{\frac{V_1^2-V_2^2}{x_1^2-x_2^2}}
B) 2 \pi \sqrt{\frac{x_1^2-x_2^2}{v_1^2-V_2^2}}
C) \frac{1}{2 \pi} \sqrt{\frac{V_2^2-V_1^2}{x_1^2-x_2^2}}
D) 2 \pi \sqrt{\frac{\mathrm{x}_1^2-\mathrm{x}_2^2}{\mathrm{~V}_2^2-\mathrm{V}_1^2}}
264
MediumMHT CET2024
For a particle executing S.H.M. having amplitude A, the speed of the article is \left(\frac{1}{3}\right)^{\text {rd }} of its maximum speed when the displacement from the mean position is
Options:
A) \frac{3 \mathrm{~A}}{\sqrt{2}}
B) \frac{2 \mathrm{~A}}{3}
C) \frac{2 \sqrt{2}}{3} \mathrm{~A}
D) \frac{\sqrt{2}}{3} \mathrm{~A}
265
MediumMHT CET2024
The motion of a particle is described by the equation a=-b x where ' a ' is the acceleration, x is the displacement from the equilibrium position and b is a constant. The periodic time will be
Options:
A) \frac{2 \pi}{\mathrm{~b}}
B) \frac{2 \pi}{\sqrt{b}}
C) 2 \pi \sqrt{b}
D) 2 \sqrt{\frac{\pi}{b}}
266
MediumMHT CET2024
A horizontal platform with a small object placed on it executes a linear S.H.M. in the vertical direction. The amplitude of oscillation is 40 cm . What should be the least period of these oscillations, so that the object is not detached from the platform? [Take \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2]
Options:
A) 0.2 \pi \mathrm{~s}
B) 0.3 \pi \mathrm{~s}
C) 0.4 \pi \mathrm{~s}
D) 0.5 \pi \mathrm{~s}
267
MediumMHT CET2024
Starting from mean position, a body oscillates simple harmonically with a period ' T '. After what time will its kinetic energy be 75 \% of the total energy? \left(\sin 30^{\circ}=0.5\right)
Options:
A) \frac{\mathrm{T}}{8}
B) \frac{\mathrm{T}}{12}
C) \frac{\mathrm{T}}{16}
D) \frac{\mathrm{T}}{24}
268
MediumMHT CET2024
The maximum velocity of a particle, executing S.H.M. with an amplitude 7 mm is 4.4 \mathrm{~ms}^{-1} The period of oscillation is \left[\pi=\frac{22}{7}\right]
Options:
A) 100 s
B) 10 s
C) 0.1 s
D) 0.01 s
269
MediumMHT CET2024
A particle is performing S.H.M. about its mean position with an amplitude ' a ' and periodic time ' T '. The speed of the particle when its displacement from mean position is \frac{a}{3} will be
Options:
A) \frac{2 \pi \mathrm{a}}{\mathrm{T}}
B) \frac{4 \sqrt{2} \pi \mathrm{a}}{3 \mathrm{~T}}
C) \frac{4 \pi^2 a}{3 \mathrm{~T}}
D) \frac{\sqrt{3} \pi^2 a}{2 \mathrm{~T}}
270
MediumMHT CET2024
A piece of wood has length, breadth and height, ' a ', ' b ' and ' c ' respectively. Its relative density, is ' d '. It is floating in water such that the side ' a ' is vertical. It is pushed down a little and released. The time period of S.H.M. executed by it is (\mathrm{g}= acceleration due to gravity)
Options:
A) 2 \pi \sqrt{\frac{a b c}{g}}
B) 2 \pi \sqrt{\frac{\mathrm{bc}}{\mathrm{dg}}}
C) 2 \pi \sqrt{\frac{g}{d a}}
D) 2 \pi \sqrt{\frac{\mathrm{ad}}{\mathrm{g}}}
271
MediumMHT CET2024
All the springs in fig. (a), (b) and (c) are identical, each having force constant K . Mass attached to each system is ' m '. If T_a, T_b and T_c are the time periods of oscillations of the three systems respectively, then
Options:
A) \mathrm{T}_{\mathrm{a}}=\sqrt{2} \mathrm{~T}_{\mathrm{b}}
B) \mathrm{T}_{\mathrm{a}}=\frac{\mathrm{T}_{\mathrm{c}}}{\sqrt{2}}
C) T_b=2 T_a
D) T_b=2 T_c
272
MediumMHT CET2024
A simple pendulum of length ' L ' has mass ' M ' and it oscillates freely with amplitude ' A '. At extreme position, its potential energy is
Options:
A) \frac{\mathrm{MgA}^2}{\mathrm{~L}}
B) \frac{2 \mathrm{MgA}^2}{\mathrm{~L}}
C) \frac{\mathrm{MgA}}{2 \mathrm{~L}}
D) \frac{\mathrm{MgA}^2}{2 \mathrm{~L}}
273
MediumMHT CET2024
A particle performing S.H.M. starts from equilibrium position and its time period is 12 second. After 2 seconds its velocity is \pi \mathrm{m} / \mathrm{s}. Amplitude of the oscillation is \left[\sin 30^{\circ}=\cos 60^{\circ}=0 \cdot 5, \sin 60^{\circ}=\cos 30^{\circ}=\sqrt{3} / 2\right]
Options:
A) 6 m
B) 12 m
C) 12 \sqrt{3} \mathrm{~m}
D) 6 \sqrt{3} \mathrm{~m}
274
MediumMHT CET2024
A particle performs linear S.H.M. at a particular instant, velocity of the particle is ' u ' and acceleration is ' \mathrm{a}_1 ' while at another instant velocity is ' V ' and acceleration is ' a_2 ' $\left(0
Options:
A) \frac{v^2-u^2}{a_1-a_2}
B) \frac{v^2+u^2}{a_1+a_2}
C) \frac{u^2+v^2}{a_1-a_2}
D) \frac{u^2-V^2}{a_1+a_2}
275
MediumMHT CET2024
A particle starts oscillating simple harmonically from its equilibrium position with time period ' T '. What is the ratio of potential energy to kinetic energy of the particle at time t=\frac{T}{12} ? $\left(\sin \left(\frac{\pi}{6}\right)=\frac{1}{2}\right)
Options:
A) 1: 2
B) 2: 1
C) 1: 3
D) 3: 1
276
MediumMHT CET2024
A particle performs linear S.H.M. When the displacement of the particle from mean position is 3 cm and 4 cm , corresponding velocities are 8 \mathrm{~cm} / \mathrm{s} and 6 \mathrm{~cm} / \mathrm{s} respectively. Its periodic time is
Options:
A) 2 \pi \mathrm{~s}
B) \pi \mathrm{s}
C) 3 \pi \mathrm{~s}
D) 4 \pi \mathrm{~s}
277
MediumMHT CET2024
A simple pendulum of length ' l ' has a brass bob attached at its lower end. It's period is ' T '. A steel bob of the same size, having density ' x ' times that of brass, replaces the brass bob. Its length is then so changed that the period becomes ' 2 T '. What is the new length?
Options:
A) 4 l x
B) \frac{4 l}{\mathrm{x}}
C) 4 l
D) 2 l
278
MediumMHT CET2024
A particle performing S.H.M. with maximum velocity ' V '. If the amplitude double and periodic time is made, \left(\frac{1}{3}\right)^{\text {rd }} then the maximum velocity is
Options:
A) 6V
B) \frac{\mathrm{V}}{3}
C) \frac{3 V}{2}
D) \frac{2 \mathrm{~V}}{3}
279
MediumMHT CET2024
Let ' l_1 ' be the length of simple pendulum. Its length changes to ' l_2 ' to increase the periodic time by 20 \%. The ratio \frac{l_2}{l_1}=
Options:
A) 1.22
B) 1.33
C) 1.44
D) 1.55
280
MediumMHT CET2024
A particle is executing a linear simple harmonic motion. Let ' \mathrm{V}_1 ' and ' \mathrm{V}_2 ' are its speed at distance ' x_1 ' and ' x_2 ' from the equilibrium position. The amplitude of oscillation is
Options:
A) \sqrt{\frac{V_1^2 x_2^2-V_2^2 x_2^2}{V_1^2-V_2^2}}
B) \sqrt{\frac{V_1^2-V_2^2}{V_1^2 x_2^2-V_2^2 x_1^2}}
C) \sqrt{\frac{V_1^2 x_2^2-V_2^2 x_1^2}{V_1^2-V_2^2}}
D) \sqrt{\frac{V_1^2 x_1^2-V_2^2 x_2^2}{V_1^2-V_2^2}}
281
MediumMHT CET2024
In S.H.M. the displacement of a particle at an instant is Y=A \cos 30^{\circ}, where A=40 \mathrm{~cm} and kinetic energy is 200 J . If force constant is 1 \times 10^{\times} \mathrm{N} / \mathrm{m}, then x will be \left(\cos 30^{\circ}=\sqrt{3} / 2\right)
Options:
A) 4
B) 3
C) 2
D) 1
282
MediumMHT CET2024
A particle is performing S.H.M. with an amplitude 4 cm . At the mean position the velocity of the particle is 12 \mathrm{~cm} / \mathrm{s}. When the speed of the particle becomes 6 \mathrm{~cm} / \mathrm{s}, the distance of the particle from mean position is
Options:
A) \sqrt{3} \mathrm{~cm}
B) \sqrt{6} \mathrm{~cm}
C) 2 \sqrt{3} \mathrm{~cm}
D) 2 \sqrt{6} \mathrm{~cm}
283
MediumMHT CET2024
The maximum velocity and maximum acceleration of a particle performing a linear S.H.M. is ' \alpha ' and ' \beta ' respectively. Then the path length of the particle is
Options:
A) \frac{\alpha^2}{\beta}
B) \frac{\beta \alpha^2}{2 \alpha^2}
C) \frac{2 \alpha^2}{\beta}
D) \frac{2 \beta}{\alpha^2}
284
MediumMHT CET2024
A mass ' m ' attached to a spring oscillates with a period of 3 second. If the mass is increased by 0.6 kg , the period increases by 3 second. The initial mass ' m ' is equal to
Options:
A) 0.1 kg
B) 0.2 kg
C) 0.3 kg
D) 0.4 kg
285
MediumMHT CET2024
The velocity of particle executing S.H.M. varies with displacement (\mathrm{x}) as 4 \mathrm{~V}^2=50-\mathrm{x}^2. The time period of oscillation is \frac{x}{7} second. The value of ' x ' is (Take \pi=\frac{22}{7})
Options:
A) 22
B) 44
C) 66
D) 88
286
MediumMHT CET2024
A simple pendulum of length l_1 has time period \mathrm{T}_1. Another simple pendulum of length l_2\left(l_1>l_2\right) has time period T_2. Then the time period of the pendulum of length \left(l_1-l_2\right) will be
Options:
A) T_1-T_2
B) \sqrt{\frac{T_1}{T_2}}
C) \sqrt{\mathrm{T}_1^2-\mathrm{T}_2^2}
D) \sqrt{\frac{T_2}{T_1}}
287
MediumMHT CET2024
Two bodies A and B of equal mass are suspended from two separate massless springs of spring constants \mathrm{K}_1 and \mathrm{K}_2 respectively. The two bodies oscillate vertically such that their maximum velocities are equal. The ratio of the amplitude of B to that of A is
Options:
A) \frac{\mathrm{K}_1}{\mathrm{~K}_2}
B) \frac{\mathrm{K}_2}{\mathrm{~K}_1}
C) \sqrt{\frac{\mathrm{K}_1}{\mathrm{~K}_2}}
D) \sqrt{\frac{\mathrm{K}_2}{\mathrm{~K}_1}}
288
MediumMHT CET2024
The period of a simple pendulum gets doubled when
Options:
A) its length is doubled.
B) its length is made four times.
C) its length is made half.
D) the mass of the bob is doubled.
289
MediumMHT CET2024
Frequency of a particle performing S.H.M. is 10 Hz . The particle is suspended from a vertical spring. At the highest point of its oscillation the spring is unstretched. Maximum speed of the particle is \left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right)
Options:
A) \frac{1}{\pi} \mathrm{~m} / \mathrm{s}
B) \frac{1}{2 \pi} \mathrm{~m} / \mathrm{s}
C) \frac{1}{4 \pi} \mathrm{~m} / \mathrm{s}
D) 2 \pi \mathrm{~m} / \mathrm{s}
290
MediumMHT CET2024
When a particle in linear S.H.M. completes two oscillations, its phase increases by
Options:
A) \pi \mathrm{~rad}.
B) 2 \pi \mathrm{~rad}.
C) 3 \pi \mathrm{~rad}.
D) 4 \pi \mathrm{~rad}.
291
MediumMHT CET2024
A small sphere oscillates simple harmonically in a watch glass whose radius of curvature is 1.6 m . The period of oscillation of the sphere in second is (acceleration due to gravity, \mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2 )
Options:
A) 0.8 \pi
B) 0.6 \pi
C) 0.4 \pi
D) 0 \cdot 2 \pi
292
MediumMHT CET2024
A tube of uniform bore of cross-sectional area ' A ' has been set up vertically with open end facing up. Now ' M ' gram of a liquid of density ' d ' is poured into it. The column of liquid in this tube will oscillate with a period ' T ', which is equal to [ g= acceleration due to gravity]
Options:
A) 2 \pi \sqrt{\frac{\mathrm{MA}}{\mathrm{gd}}}
B) 2 \pi \sqrt{\frac{\mathrm{M}}{2 \mathrm{Adg}}}
C) 2 \pi \sqrt{\frac{M}{g}}
D) 2 \pi \sqrt{\frac{M}{g d A}}
293
MediumMHT CET2024
A spring has a certain mass suspended from it and its period of vertical oscillations is T_1. The spring is now cut into two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillations is now \mathrm{T}_2. The ratio of T_2 / T_1 is
Options:
A) 1: 2
B) 1: \sqrt{2}
C) \sqrt{2}: 1
D) 2: 1
294
MediumMHT CET2024
For a body performing simple harmonic motion, its potential energy is \mathrm{E}_{\mathrm{x}} at displacement x and \mathrm{E}_{\mathrm{y}} at displacement y from mean position. The potential energy E_0 at displacement (x+y) is
Options:
A) \sqrt{\mathrm{E}_{\mathrm{x}}^2+\mathrm{E}_{\mathrm{y}}^2}
B) \sqrt{E_x-E_y}
C) E_x+E_y
D) E_x+E_y+2 \sqrt{E_x E_y}
295
MediumMHT CET2024
The displacement of a particle performing S.H.M. is given by Y=A \cos [\pi(t+\phi)]. If at \mathrm{t}=0, the displacement is \mathrm{y}=2 \mathrm{~cm} and velocity is 2 \pi \mathrm{~cm} / \mathrm{s}, the value of amplitude A in cm is
Options:
A) 2
B) \sqrt{2}
C) 2 \sqrt{2}
D) \frac{1}{\sqrt{2}}
296
MediumMHT CET2024
A particle is performing simple harmonic motion and if the oscillations are Camped oscillations then the angular frequency is given by
Options:
A) \sqrt{\frac{k}{m}+\left(\frac{b}{2 m}\right)^2}
B) \frac{\mathrm{k}}{\mathrm{m}}+\left(\frac{\mathrm{b}}{2 \mathrm{~m}}\right)^2
C) \sqrt{\frac{k}{m}-\left(\frac{b}{2 m}\right)^2}
D) \frac{\mathrm{k}}{\mathrm{m}}-\left(\frac{\mathrm{b}}{2 \mathrm{~m}}\right)^2
297
MediumMHT CET2024
Choose the correct answer. When a point of suspension of pendulum is moved vertically upward with acceleration ' a ', its period of oscillation
Options:
A) decreases
B) increases
C) remains same
D) some times increases and some times decreases
298
MediumMHT CET2024
Three masses 500 \mathrm{~g}, 300 \mathrm{~g} and 100 g are suspended at the end of spring as shown in figure and are in equilibrium. When the 500 g mass is removed, the system oscillates with a period of 3 second. When the 300 g mass is also removed it will oscillate with a period of
Options:
A) 1 s
B) 1.5 s
C) 2 s
D) 2.5 s
299
MediumMHT CET2023
A uniform circular disc of mass $12 \mathrm{~kg} is held by two identical springs. When the disc is slightly pressed down and released, it executes S.H.M. of period 2 second. The force constant of each spring is (nearly) (Take \pi^2=10$ )
Options:
A) 230 Nm$^{-1}
B) 120 Nm$^{-1}
C) 60 Nm$^{-1}
D) 30 Nm$^{-1}
300
MediumMHT CET2023
A light spring is suspended with mass $m_1 at its lower end and its upper end fixed to a rigid support. The mass is pulled down a short distance and then released. The period of oscillation is T second. When a mass m_2 is added to m_1 and the system is made to oscillate, the period is found to be \frac{3}{2} T. The ratio m_1: m_2$ is
Options:
A) 2: 3
B) 3: 4
C) 4: 5
D) 5: 6
301
MediumMHT CET2023
A block of mass '$M' rests on a piston executing S.H.M. of period one second. The amplitude of oscillations, so that the mass is separated from the piston, is (acceleration due to gravity, \mathrm{g}=10 \mathrm{~ms}^{-2}, \pi^2=10$ )
Options:
A) 0.25 m
B) 0.5 m
C) 1 m
D) \infty
302
MediumMHT CET2023
A simple pendulum of length '$l' and a bob of mass '\mathrm{m}' is executing S.H.M. of small amplitude 'A'. The maximum tension in the string will be (\mathrm{g}=$ acceleration due to gravity)
Options:
A) 2 \mathrm{~mg}
B) \mathrm{mg}\left[1+\left(\frac{\mathrm{A}}{\ell}\right)^2\right]
C) \mathrm{mg}\left[1+\left(\frac{\mathrm{A}}{\ell}\right)\right]^2
D) m g\left[1+\left(\frac{\mathrm{A}}{\ell}\right)\right]
303
MediumMHT CET2023
The displacement of a particle executing S.H.M. is $x=\mathrm{a} \sin (\omega t-\phi). Velocity of the particle at time \mathrm{t}=\frac{\phi}{\omega} is \left(\cos 0^{\circ}=1\right)
Options:
A) \omega \cos \phi
B) \mathrm{a} \omega
C) \omega \cos 2 \phi
D) -\mathrm{a} \omega \cos 2 \phi
304
MediumMHT CET2023
The bob of simple pendulum of length '$L' is released from a position of small angular displacement \theta. Its linear displacement at time '\mathrm{t}' is ( \mathrm{g}=$ acceleration due to gravity)
Options:
A) L \theta \cos \left[\sqrt{\frac{g}{L}} \cdot t\right]
B) L \theta \sin \left[2 \pi \sqrt{\frac{g}{L}} \cdot t\right]
C) L \theta \cos \left[2 \pi \sqrt{\frac{g}{L}} \cdot t\right]
D) L \theta \sin \left[\sqrt{\frac{g}{L}} \cdot t\right]
305
MediumMHT CET2023
Under the influence of force $F_1 the body oscillates with a period T_1 and due to another force F_2 body oscillates with period T_2$. If both forces acts simultaneously, then the resultant period is (consider displacement is same in all three cases)
Options:
A) T=\sqrt{\frac{T_1^2+T_2^2}{T_1^2 T_2^2}}
B) T=\sqrt{\frac{T_1^2 T_2^2}{T_1^2+T_2^2}}
C) T=\sqrt{\frac{T_1^2}{T_2^2}}
D) T=\sqrt{T_1^2+T_2^2}
306
MediumMHT CET2023
A mass $M is suspended from a light spring. An additional mass M_1 added extends the spring further by a distance x. Now, the combined mass will oscillate on the spring with period T=
Options:
A) 2 \pi\left[\left(\frac{M_1 g}{x\left(M+M_1\right)}\right)\right]^{\frac{1}{2}}
B) 2 \pi\left[\frac{\left(M+M_1\right) x}{M_1 g}\right]^{\frac{1}{2}}
C) \left(\frac{\pi}{2}\right)\left[\left(\frac{M_1 g}{x\left(M+M_1\right)}\right)\right]^{\frac{1}{2}}
D) 2 \pi\left[\left(\frac{M+M_1}{M_1 g x}\right)\right]^{\frac{1}{2}}
307
MediumMHT CET2023
A body of mass $0.04 \mathrm{~kg} executes simple harmonic motion (SHM) about \mathrm{x}=0 under the influence of force \mathrm{F}$ as shown in graph. The period of
Options:
A) 2 \pi \mathrm{s}
B) 0.2 \pi \mathrm{s}
C) \pi \mathrm{d}
D) \frac{\pi}{2} \mathrm{~s}
308
MediumMHT CET2023
A simple pendulum has a time period '$T$' in air. Its time period when it is completely immersed in a liquid of density one eighth the density of the material of bob is
Options:
A) \left(\sqrt{\frac{7}{8}}\right) \mathrm{T}
B) \left(\sqrt{\frac{5}{8}}\right) \mathrm{T}
C) \left(\sqrt{\frac{3}{8}}\right) \mathrm{T}
D) \left(\sqrt{\frac{8}{7}}\right) \mathrm{T}
309
MediumMHT CET2023
In a stationary lift, time period of a simple pendulum is '$\mathrm{T}'. The lift starts accelerating downwards with acceleration \left(\frac{\mathrm{g}}{4}\right)$, then the time period of the pendulum will be
Options:
A) \frac{\sqrt{3}}{2} \mathrm{~T}
B) \frac{2}{\sqrt{3}} \mathrm{~T}
C) \frac{3}{4} \mathrm{~T}
D) \frac{4}{3} \mathrm{~T}
310
MediumMHT CET2023
A particle starts from mean position and performs S.H.M. with period 4 second. At what time its kinetic energy is $50 \% of total energy? \left(\cos 45^{\circ}=\frac{1}{\sqrt{2}}\right)
Options:
A) 0.1 s
B) 0.2 s
C) 0.4 s
D) 0.5 s
311
MediumMHT CET2023
A simple pendulum performs simple harmonic motion about $\mathrm{x}=0 with an amplitude '\mathrm{a}' and time period 'T'. The speed of the pendulum at x=\frac{a}{2}$ is
Options:
A) \frac{\pi \mathrm{a}}{\mathrm{T}}
B) \frac{3 \pi^2 \mathrm{a}}{\mathrm{T}}
C) \frac{\pi \mathrm{a} \sqrt{3}}{\mathrm{~T}}
D) \frac{\pi \mathrm{a} \sqrt{3}}{2}
312
MediumMHT CET2023
Four massless springs whose force constants are $2 \mathrm{~K}, 2 \mathrm{~K}, \mathrm{~K} and 2 \mathrm{~K} respectively are attached to a mass \mathrm{M} kept on a frictionless plane as shown in figure, If mass M$ is displaced in horizontal direction then frequency of oscillating system is
Options:
A) \frac{1}{2 \pi} \sqrt{\frac{\mathrm{K}}{4 \mathrm{M}}}
B) \frac{1}{2 \pi} \sqrt{\frac{4 \mathrm{~K}}{\mathrm{M}}}
C) \frac{1}{2 \pi} \sqrt{\frac{\mathrm{K}}{7 \mathrm{M}}}
D) \frac{1}{2 \pi} \sqrt{\frac{7 \mathrm{~K}}{\mathrm{M}}}
313
MediumMHT CET2023
The upper end of the spring is fixed and a mass '$m' is attached to its lower end. When mass is slightly pulled down and released, it oscillates with time period 3 second. If mass '\mathrm{m}' is increased by 1 \mathrm{~kg}, the time period becomes 5 second. The value of '\mathrm{m}$' is (mass of spring is negligible)
Options:
A) \frac{3}{8} \mathrm{~kg}
B) \frac{5}{9} \mathrm{~kg}
C) \frac{8}{13} \mathrm{~kg}
D) \frac{9}{16} \mathrm{~kg}
314
MediumMHT CET2023
For a particle executing S.H.M., its potential energy is 8 times its kinetic energy at certain displacement '$x' from the mean position. If 'A' is the amplitude of S.H.M the value of 'x$' is
Options:
A) \frac{\mathrm{A} \sqrt{2}}{3}
B) \mathrm{A} \sqrt{3}
C) \frac{2 \sqrt{2} \mathrm{~A}}{3}
D) \frac{\mathrm{A}}{\sqrt{2}}
315
MediumMHT CET2023
The time period of a simple pendulum inside a stationary lift is '$T'. When the lift starts accelerating upwards with an acceleration \left(\frac{\mathrm{g}}{3}\right)$, the time period of the pendulum will be
Options:
A) \frac{\sqrt{5}}{2} \mathrm{~T}
B) \frac{\sqrt{3}}{2} \mathrm{~T}
C) \frac{2 \mathrm{~T}}{\sqrt{3}}
D) \frac{2 \mathrm{~T}}{\sqrt{5}}
316
MediumMHT CET2023
A body is executing a linear S.H.M. Its potential energies at the displacement '$\mathrm{x}' and '\mathrm{y}' are '\mathrm{E}_1' and 'E_2' respectively. Its potential energy at displacement (\mathrm{x}+\mathrm{y})$ will be
Options:
A) \mathrm{E}_1+\mathrm{E}_2
B) \left(\sqrt{\mathrm{E}_1}+\sqrt{\mathrm{E}_2}\right)^2
C) \quad \mathrm{E}_1-\mathrm{E}_2
D) \left(\sqrt{\mathrm{E}_2}-\sqrt{\mathrm{E}_1}\right)^2
317
MediumMHT CET2023
A simple harmonic progressive wave is represented by $y=A \sin (100 \pi t+3 x). The distance between two points on the wave at a phase difference of \frac{\pi}{3}$ radian is
Options:
A) \frac{\pi}{8} \mathrm{~m}
B) \frac{\pi}{9} \mathrm{~m}
C) \frac{\pi}{6} \mathrm{~m}
D) \frac{\pi}{3} \mathrm{~m}
318
MediumMHT CET2023
The amplitude of a particle executing S.H.M. is $3 \mathrm{~cm}. The displacement at which its kinetic energy will be 25 \%$ more than the potential energy is
Options:
A) 1 \mathrm{~cm}
B) 2 \mathrm{~cm}
C) 3 \mathrm{~cm}
D) 4 \mathrm{~cm}
319
MediumMHT CET2023
Two S.H.Ms. are represented by equations $\mathrm{y}_1=0.1 \sin \left(100 \pi \mathrm{t}+\frac{\pi}{3}\right) and \mathrm{y}_2=0.1 \cos (100 \pi \mathrm{t})$ The phase difference between the speeds of the two particles is
Options:
A) \frac{\pi}{3}
B) -\frac{\pi}{6}
C) +\frac{\pi}{6}
D) -\frac{\pi}{3}
320
MediumMHT CET2023
A spring has a certain mass suspended from it and its period for vertical oscillations is '$T_1'. The spring is now cut in to two equal halves and the same mass is suspended from one of the halves. The period of vertical oscillations is now '\mathrm{T}_2'. The ratio \mathrm{T}_1 / \mathrm{T}_2$ is
Options:
A) 2
B) \sqrt{2}
C) \frac{1}{\sqrt{2}}
D) \frac{1}{2}
321
MediumMHT CET2023
A particle is vibrating in S.H.M. with an amplitude of $4 \mathrm{~cm}$. At what displacement from the equilibrium position is its energy half potential and half kinetic?
Options:
A) 1 \mathrm{~cm}
B) \sqrt{2} \mathrm{~cm}
C) 2 \mathrm{~cm}
D) 2 \sqrt{2} \mathrm{~cm}
322
MediumMHT CET2023
A rubber ball filled with water, having a small hole is used as the bob of a simple pendulum. The time period of such a pendulum
Options:
A) is a constant.
B) decreases with time.
C) increases with time.
D) first increases and then decreases, finally having same value as at the beginning.
323
MediumMHT CET2023
The maximum velocity of a particle performing S.H.M. is '$\mathrm{V}'. If the periodic time is made \left(\frac{1}{3}\right)^d$ and the amplitude is doubled, then the new maximum velocity of the particle will be
Options:
A) \frac{\mathrm{V}}{6}
B) \frac{3 \mathrm{~V}}{2}
C) 3 \mathrm{~V}
D) 6 \mathrm{~V}
324
MediumMHT CET2022
The time taken by a particle executing simple harmonic motion of period '$\mathrm{T}$', to move from the mean position to half the maximum displacement is
Options:
A) \frac{\mathrm{T}}{12} \mathrm{~s}
B) \frac{\mathrm{T}}{2} \mathrm{~s}
C) \frac{\mathrm{T}}{4} \mathrm{~s}
D) \frac{\mathrm{T}}{6} \mathrm{~s}
325
MediumMHT CET2022
In a medium, the phase difference between two particles separated by a distance '$x' is \left(\frac{\pi}{5}\right)^{\text {c }}. If the frequency of the oscillation of particles is 25 \mathrm{~Hz} and the velocity of propagation of the waves is 75 \mathrm{~m} / \mathrm{s}, then the value of x$ is
Options:
A) 0.4 m
B) 0.1 m
C) 0.2 m
D) 0.3 m
326
MediumMHT CET2022
A particle starts oscillating simple harmonically from its mean position with time period '$T'. At time t=\frac{T}{12}, the ratio of the potential energy to kinetic energy of the particle is \left(\sin 30^{\circ}=\cos 60^{\circ}=0.5, \cos 30^{\circ}=\sin 60^{\circ}=\sqrt{3} / 2\right)
Options:
A) 1: 2
B) 3: 1
C) 2: 1
D) 1: 3
327
MediumMHT CET2021
The displacement of a particle performing S.H.M. is given by $x=5 \sin (3 t+3), where x is in \mathrm{cm} and t$ is in second. The maximum acceleration of the particle will be
Options:
A) 15 \mathrm{~cm} \mathrm{~s}^{-2}
B) 30 \mathrm{~cm} \mathrm{~s}^{-2}
C) 45 \mathrm{~cm} \mathrm{~s}^{-2}
D) 90 \mathrm{~cm} \mathrm{~s}^{-2}
328
MediumMHT CET2021
Two identical springs of constant '$\mathrm{K}' are connected in series and parallel in shown in figure. A mass '\mathrm{M}$' is suspended from them. The ratio of their frequencies is series to parallel combination will be
Options:
A) 1: 2
B) 1: 4
C) 4: 1
D) 1: \sqrt{2}
329
MediumMHT CET2021
A particle performing linear S.H.M. of amplitude $0.1 \mathrm{~m} has displacement 0.02 \mathrm{~m} and acceleration 0.5 \mathrm{~m} / \mathrm{s}^2. The maximum velocity of the particle in \mathrm{m} / \mathrm{s}$ is
Options:
A) 0.05
B) 0.50
C) 0.01
D) 0.25
330
MediumMHT CET2021
A body is executing S.H.M. under the action of force having maximum magntude $50 \mathrm{~N}$. When its energy is half kinetic and half potential; the magnitude of the force acting on the particle is
Options:
A) \frac{25}{\sqrt{2}} \mathrm{~N}
B) 50 \mathrm{~N}
C) 25 \mathrm{~N}
D) 25 \sqrt{2} \mathrm{~N}
331
MediumMHT CET2021
A bob of simple pendulum of mass 'm' perform $\mathrm{SHM} with amplitude '\mathrm{A}' and period 'T'. Kinetic energy of pendulum of displacement x=\frac{A}{2}$ will be
Options:
A) \frac{2 \mathrm{~m} \pi^2 A}{3 \mathrm{~T}^2}
B) \mathrm{\frac{3 m \pi^2 A}{2 T}}
C) \frac{2 \mathrm{~m} \pi \mathrm{A}^2}{3 \mathrm{~T}}
D) \frac{2 \mathrm{~m} \pi^2 \mathrm{~A}^2}{2 \mathrm{~T}^2}
332
MediumMHT CET2021
An object executes SHM along $x-axis with amplitude 0.06 \mathrm{~m}. At certain distance '\mathrm{x}' metre from mean position, it has kinetic energy 10 \mathrm{~J} and potential energy 8 \mathrm{~J}. the distance '\mathrm{x}$' will be
Options:
A) 0.08 m
B) 0.02 m
C) 0.04 m
D) 0.06 m
333
MediumMHT CET2021
A body executes SHM under the action of force '$\mathrm{F}_1' with time period '\mathrm{T}_1'. If the force is changed to '\mathrm{F_2}', it executes SHM with period '\mathrm{T_2}'. If both the forces '\mathrm{F_1}' and '\mathrm{F}_2$' act simultaneously in the same direction on the body, its time period is
Options:
A) \frac{\sqrt{\mathrm{T}_1^2-\mathrm{T}_2^2}}{\mathrm{~T}_1 \mathrm{~T}_2}
B) \frac{T_1 T_2}{\sqrt{T_1^2-T_2^2}}
C) \frac{\sqrt{T_1^2+\mathrm{T}_2^2}}{\mathrm{~T}_1 \mathrm{~T}_2}
D) \frac{\mathrm{T}_1 \mathrm{~T}_2}{\sqrt{\mathrm{T}_1^2+\mathrm{T}_2^2}}
334
MediumMHT CET2021
A particle performing S.H.M. when displacement is '$x', the potential energy and restoring force acting on it are denoted by 'E' and 'F' respectively. The relation between x, E and F$ is
Options:
A) \frac{2 E}{F}-x^2=0
B) \frac{2 \mathrm{E}}{\mathrm{F}}+\mathrm{x}^2=0
C) \frac{2 E}{F}+x=0
D) \frac{2 E}{F}-x=0
335
MediumMHT CET2021
A body is performing S.H.M. of amplitude 'A'. The displacement of the body from a point where kinetic energy is maximum to a point where potential energy is maximum, is
Options:
A) zero
B) \pm \mathrm{A}
C) \pm \frac{\mathrm{A}}{2}
D) \pm \frac{\mathrm{A}}{4}
336
MediumMHT CET2021
A particle excuting S.H.M starts from the mean position. Its amplitude is 'A' and time period '$\mathrm{T}$' At what displacement its speed is one-fourth of the maximum speed?
Options:
A) \frac{\mathrm{A}}{\sqrt{15}}
B) \frac{\mathrm{A}}{4}
C) \frac{4 \mathrm{~A}}{15}
D) \frac{\mathrm{A} \sqrt{15}}{40}
337
MediumMHT CET2021
A particle connected to the end of a spring executes S.H.M. with period '$T_1'. While the corresponding period for another spring is '\mathrm{T}_2$'. If the period of oscillation with two springs in series is 'T', then
Options:
A) \mathrm{T}=\sqrt{\mathrm{T}_1^2+\mathrm{T}_2^2}
B) \mathrm{T}=\sqrt{\mathrm{T}_2^2-\mathrm{T}_1^2}
C) \mathrm{T=T_1+T_2}
D) \mathrm{T}=\mathrm{T}_1-\mathrm{T}_2
338
MediumMHT CET2021
'$n$' waves are produced on a string in 1 second. When the radius of the string is doubled, keeping tension same, the number of waves produced in 1 second for the same harmonic will be
Options:
A) 2 \mathrm{n}
B) \frac{\mathrm{n}}{2}
C) \frac{\mathrm{n}}{\sqrt{2}}
D) \sqrt{2} \mathrm{n}
339
MediumMHT CET2021
A body of mass '$m' performs linear S.H.M. given by equation x=P \sin \omega t+Q \sin \left(\omega t+\frac{\pi}{2}\right)$. The total energy of the particle at any instant is
Options:
A) \frac{1}{2} \mathrm{~m} \omega^2 \mathrm{PQ}
B) \frac{1}{2} \frac{\mathrm{m} \omega^2}{\mathrm{P}^2 \mathrm{Q}^2}
C) \frac{1}{2} m \omega^2\left(P^2+Q^2\right)
D) \frac{1}{2} m \omega^2 p^2 Q^2
340
MediumMHT CET2021
A mass $0.4 \mathrm{~kg} performs S.H.M. with a frequency \frac{16}{\pi} \mathrm{Hz}. At a certain displacement it has kinetic energy 2 \mathrm{~J} and potential energy 1.2 \mathrm{~J}$. The amplitude of oscillation is
Options:
A) 0.15 \mathrm{~m}
B) 0.125 \mathrm{~m}
C) 0.075 \mathrm{~m}
D) 0.1 \mathrm{~m}
341
MediumMHT CET2021
If the amplitude of linear S.H.M. is decreased then
Options:
A) its period and total energy will increase.
B) its period will increase and total energy will decrease
C) its period and total energy will decrease.
D) its period will not change but total energy will decrease
342
MediumMHT CET2021
A child is sitting on a swing which performs S.H.M. It has minimum and maximum heights from ground $0.75 \mathrm{~cm} and 2 \mathrm{~m} respectively. Its maximum speed will be \left[\mathrm{g}=10 \frac{\mathrm{m}}{\mathrm{s}^2}\right]
Options:
A) \sqrt{1.25} \mathrm{~m} / \mathrm{s}
B) \sqrt{12.5} \mathrm{~m} / \mathrm{s}
C) 5 \mathrm{~m} / \mathrm{s}
D) 25 \mathrm{~m} / \mathrm{s}
343
MediumMHT CET2021
A pendulum clock is running fast. To correct its time, we should
Options:
A) reduce the mass of the bob.
B) reduce the amplitude of oscillation.
C) increase the length of the pendulum.
D) reduce the length of the pendulum.
344
MediumMHT CET2021
A particle is performing S.H.M. with maximum velocity '$v$'. If the amplitude is tripled and periodic time is doubled then maximum velocity will be
Options:
A) 1.5 \mathrm{~v}
B) 3 \mathrm{~v}
C) 2 \mathrm{~v}
D) \mathrm{v}
345
MediumMHT CET2021
A particle executes S.H.M. of period $\frac{2 \pi}{\sqrt{3}} second along a straight line 4 \mathrm{~cm}$ long. The displacement of the particle at which the velocity is numerically equal to the acceleration is
Options:
A) 2 \mathrm{~cm}
B) 1 \mathrm{~cm}
C) 4 \mathrm{~cm}
D) 3 \mathrm{~cm}
346
MediumMHT CET2021
A particle is suspended from a vertical spring which is executing S.H.M. of frequency $5 \mathrm{~Hz}. The spring is unstretched at the highest point of oscillation. Maximum speed of the particle is (\mathrm{g} =10 \mathrm{~m} / \mathrm{s}^2)
Options:
A) \frac{1}{\pi} \mathrm{m} / \mathrm{s}
B) \frac{1}{4 \pi} \mathrm{m} / \mathrm{s}
C) \frac{1}{2 \pi} \mathrm{m} / \mathrm{s}
D) \pi~ \mathrm{m} / \mathrm{s}
347
MediumMHT CET2021
A body performs S.H.M. under the action of force '$\mathrm{F}_1' with period '\mathrm{T}_1' second. If the force is changed to '\mathrm{F}_2' it performs S.H.M. with period '\mathrm{T_2}' second. If both forces '\mathrm{F_1}' and '\mathrm{F_2}$' act simultaneously in the same direction on the body, the period in second will be
Options:
A) \frac{T_1+T_2}{T_1 T_2}
B) \frac{T_1^2+T_2^2}{T_1 T_2}
C) \frac{\mathrm{T}_1 \mathrm{~T}_2}{\sqrt{\mathrm{T}_1^2+\mathrm{T}_2^2}}
D) \frac{T_1 T_2}{T_1+T_2}
348
MediumMHT CET2021
A mass '$\mathrm{m}_1' is suspended from a spring of negligible mass. A spring is pulled slightly in downward direction and released, mass performs S.H.M. of period '\mathrm{T}_1'. If the mass is increased by '\mathrm{m}_2', the time period becomes '\mathrm{T}_2'. The ratio \frac{\mathrm{m}_2}{\mathrm{~m}_1}$ is
Options:
A) \frac{\mathrm{T}_1^2+\mathrm{T}_2^2}{\mathrm{~T}_1^2}
B) \frac{\mathrm{T}_1-\mathrm{T}_2}{\mathrm{~T}_1}
C) \frac{\mathrm{T}_2^2-\mathrm{T}_1^2}{\mathrm{~T}_1^2}
D) \frac{T_1^2-T_2^2}{T_1^2}
349
MediumMHT CET2021
Two particles $\mathrm{P} and \mathrm{Q} performs S.H.M. of same amplitude and frequency along the same straight line. At a particular instant, maximum distance between two particles is \sqrt{2} a. The initial phase difference between them is \left[\sin ^{-1}\left(\frac{1}{\sqrt{2}}\right)=\cos ^{-1}\left(\frac{1}{\sqrt{2}}\right)=\frac{\pi}{4}\right]
Options:
A) \frac{\pi}{6}
B) \frac{\pi}{2}
C) zero
D) \frac{\pi}{3}
350
MediumMHT CET2021
A particle of mass 5kg is executing S.H.M. with an amplitude 0.3 m and time period $\frac{\pi}{5}$s. The maximum value of the force acting on the particle is
Options:
A) 0.15 N
B) 4 N
C) 5 N
D) 0.3 N
351
MediumMHT CET2020
Two bodies A and B of equal mass are suspended from two separate massles springs of force constant k_1 and k_2, respectively. The bodies oscillate vertically such that their maximum velocities are equal. The ratio of the amplitudes of body A to that of body B is
Options:
A) \sqrt{\frac{k_2}{k_1}}
B) \frac{k_2}{k_1}
C) \frac{k_1}{k_2}
D) \sqrt{\frac{k_1}{k_2}}
352
MediumMHT CET2020
A bob of a simple pendulum has mass m and is oscillating with an amplitude a. If the length of the pendulum is L, then the maximum tension in the string is \left[\cos 0^{\circ}=1\right., g= acceleration due to gravity]
Options:
A) m g\left[1+\left(\frac{a}{L}\right)^2\right]
B) m g\left[1-\left(\frac{L}{a}\right)^2\right]
C) m g\left[1+\left(\frac{L}{a}\right)^2\right]
D) m g\left[1-\left(\frac{a}{L}\right)^2\right]
353
MediumMHT CET2020
A body of mass 64 g is made to oscillate turn by turn on two different springs A and B. Spring A and B has force constant 4 \frac{\mathrm{~N}}{\mathrm{~m}} and 16 \frac{\mathrm{~N}}{\mathrm{~m}} respectively. If T_1 and T_2 are period of oscillations of springs A and B respectively, then \frac{T_1+T_2}{T_1-T_2} will be
Options:
A) 1: 2
B) 1: 3
C) 3: 1
D) 2: 1
354
MediumMHT CET2020
The damping force of an oscillator is directly proportional to the velocity. The unit of constant of proportionality is
Options:
A) \mathrm{kg}~ \mathrm{ms}^{-1}
B) \mathrm{kg} ~\mathrm{s}^{-1}
C) \mathrm{kg} ~\mathrm{ms}^{-2}
D) kg-s
355
MediumMHT CET2020
A particle performs simple harmonic motion with period of 3 s . The time taken by it to cover a distance equal to half the amplitude from mean position is [$\sin 30^{\circ}=0.5$]
Options:
A) \frac{1}{4} \mathrm{~s}
B) \frac{3}{2} \mathrm{~s}
C) \frac{3}{4} \mathrm{~s}
D) \frac{1}{2} \mathrm{~s}
356
MediumMHT CET2020
A simple pendulum of length $I has a bob of mass m. It executes SHM of small amplitude A. The maximum tension in the string is (g=$ acceleration due to gravity)
Options:
A) m g
B) m g\left(\frac{A^2}{I^2}+1\right)
C) 2 mg
D) m g\left(\frac{A}{I}+1\right)
357
MediumMHT CET2020
A block of mass $m attached to one end of the vertical spring produces extension x$. If the block is pulled and released, the periodic time of oscillation is
Options:
A) 2 \pi \sqrt{\frac{x}{4 g}}
B) 2 \pi \sqrt{\frac{2 x}{g}}
C) 2 \pi \sqrt{\frac{x}{2 g}}
D) 2 \pi \sqrt{\frac{x}{g}}
358
MediumMHT CET2020
A simple pendulum of length $L has mass m and it oscillates freely with amplitude A. At extreme position, its potential energy is (g=$ acceleration due to gravity)
Options:
A) \frac{m g A}{L}
B) \frac{m g A}{2 l}
C) \frac{m g A^2}{L}
D) \frac{m g A^2}{2 L}
359
MediumMHT CET2020
For a particle performing SHM when displacement is $x, the potential energy and restoring force acting on it is denoted by E and F, respectively. The relation between x, E and F$ is
Options:
A) \frac{E}{F}+x=0
B) \frac{2 E}{F}-x=0
C) \frac{2 E}{F}+x=0
D) \frac{E}{F}-x=0
360
MediumMHT CET2019
A person measures a time period of a simple pendulum inside a stationary lift and finds it to be T. If the lift starts accelerating upwards with an acceleration \left(\frac{g}{3}\right), the time period of the pendulum will be
Options:
A) \frac{T}{\sqrt{3}}
B) \sqrt{3} \frac{T}{2}
C) \sqrt{3} T
D) \frac{T}{3}
361
MediumMHT CET2019
In damped SHM, the SI unit of damping constant is
Options:
A) \frac{\mathrm{N}}{\mathrm{s}}
B) \frac{\mathrm{kg}}{\mathrm{s}}
C) \frac{\mathrm{kg}}{\mathrm{m}}
D) \frac{N}{m}
362
MediumMHT CET2019
The total energy of a simple harmonic oscillaior is proportional to
Options:
A) square of this ampitude
B) square root of displacement
C) amplitude
D) frequency
363
MediumMHT CET2019
A particle is performing a linear simple harmonic motion of amplitude ' A '. When it is midway between its mean and extreme position, the magnitudes of its velocity and acceleration are equal. What is the periodic time of the motion?
Options:
A) \frac{2 \pi}{\sqrt{3}} \mathrm{~s}
B) \frac{\sqrt{3}}{2 \pi} s
C) 2 \pi \sqrt{3} \mathrm{~s}
D) \frac{1}{2 \pi \sqrt{3}} \mathrm{~s}
364
MediumMHT CET2019
Two pendulums begin to swing simultaneously. The first pendulum makes nine full oscillations when the other makes seven. The ratio of the lengths of the two pendulums is
Options:
A) \frac{49}{81}
B) \frac{64}{81}
C) \frac{8}{9}
D) \frac{7}{9}
365
MediumMHT CET2019
If ' x ', v ' and ' a ' denote the displacement, velocity and acceleration of a particle respectively executing SHM of periodic time h then which one of the following does not change with time?
Options:
A) \frac{a T}{x}
B) a t+2 \pi v
C) \frac{a T}{v}
D) a T+4 \pi^2 v^2
366
MediumMHT CET2019
The quantity which does not vary periodically for a particle performing SHM is
Options:
A) acceleration
B) total energy
C) displacement
D) velocity
367
MediumMHT CET2019
A particle executes the simple harmonic motion with an amplitude ' A '. The distance travelled by it in one periodic time is
Options:
A) \frac{A}{2}
B) A
C) 2 A
D) 4 A
368
MediumVITEEE2024
The potential energy of a particle of mass 0.1 kg moving along the X-axis, is given by U=5 x(x-4) \mathrm{J}, where x is in metres. Choose the wrong option.
Options:
A) The speed of the particle is maximum at x=2 \mathrm{~m}
B) The particle executes simple harmonic motion
C) The period of oscillation of the particle is \frac{\pi}{5} \mathrm{~s}
D) None of the above
369
MediumVITEEE2022
One end of a spring of force constant $k is fixed to a vertical wall and other to a body of mass m resting on a smooth horizontal surface. There is another wall at a distance x_0 from the body. The spring is then compressed by 2 x_0$ and released. The time taken to stike the wall is
Options:
A) \frac{\pi}{6} \sqrt{\frac{m}{k}}
B) \sqrt{\frac{m}{k}}
C) \frac{2 \pi}{3} \sqrt{\frac{m}{k}}
D) \frac{\pi}{4} \sqrt{\frac{m}{k}}
369
Total Questions
70
Easy
294
Medium
5
Hard
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