Question 51: Oscillations

Question

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 $\pi = \frac{22}{7}$ ]

Accepted Answer

Let the displacement $(\mathrm{x})$ of a simple harmonic oscillator at any time t be represented by :
$$ x=A \sin (\omega t) $$
Where :
A is the amplitude.
$\omega$ is the angular frequency of the oscillator.
The velocity ( $v$ ) is the derivative of displacement with respect to time :
$$ v=\frac{d x}{d t}=A \omega \cos (\omega t) $$
The kinetic energy (K.E.) of the oscillator is given by :
$$ 	ext { K. } \mathrm{E}=\frac{1}{2} \mathrm{~m} \mathrm{v}^2 $$
$$\Rightarrow $$ K. $\mathrm{E} .=\frac{1}{2} \mathrm{~m}(\mathrm{~A} \omega \cos (\omega \mathrm{t}))^2$
$$\Rightarrow $$ K. $\mathrm{E} .=\frac{1}{2} \mathrm{~m} \mathrm{~A}^2 \omega^2 \cos ^2(\omega \mathrm{t})$
$$\Rightarrow $$ K. $E \cdot=\frac{1}{2} m A^2 \omega^2\left[\frac{1+\cos (2 \omega t)}{2}ight]$
$$\Rightarrow $$ K. $E \cdot=\frac{1}{4} m A^2 \omega^2+\frac{1}{4} m A^2 \omega^2 \cos (2 \omega t)$
$$\Rightarrow $$ $$ 	ext { K. } \mathrm{E} .=\frac{1}{4} \mathrm{~m} \mathrm{~A}^2 \omega^2+\frac{1}{4} \mathrm{~m} \mathrm{~A}^2 \omega^2 \cos \left(\omega^{\prime} \mathrm{t}ight) $$
Here, $\omega^{\prime}=2 \omega$ is the frequency of oscillation of kinetic energy of oscillator.
So, the kinetic energy oscillates with an angular frequency of $2 \omega$.
The kinetic energy oscillates with an angular frequency $\omega^{\prime}=176 \mathrm{rad} / \mathrm{s}$.
$$ \Rightarrow \omega=\frac{176}{2}=88 \mathrm{rad} / \mathrm{s} $$
Using the relationship between angular frequency ( $\omega$ ) and frequency (f) in Hz is:
$$ \omega=2 \pi \mathrm{f} $$
$$\Rightarrow $$ $$ f=\frac{\omega}{2 \pi} $$
Substitute the values $\left(\omega=88ight.$ and $\left.\pi=\frac{22}{7}ight)$ :
$$ \mathrm{f}=\frac{88}{2 	imes 22 / 7} $$
$$\Rightarrow $$ $f=\frac{88 	imes 7}{44}$
$$\Rightarrow $$ $\mathrm{f}=14 \mathrm{~Hz}$

Answer

88

Answer

14

Answer

28

Answer

176

Oscillations

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} ]

Choose the correct answer: