Question 58: Oscillations

Question

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 :

Accepted Answer

$$\begin{aligned}
& \mathrm{x}=\mathrm{A} \sin (\omega \mathrm{t}+\phi) \
& \mathrm{x}_0=\mathrm{A} \sin \phi \quad	ext{.... (1)}\
& \mathrm{p}=\mathrm{mA} \omega \cos (\omega \mathrm{t}+\phi) \
& \mathrm{p}_0=\mathrm{mA} \omega \cos \phi \quad	ext{.... (2)}
\end{aligned}$$
$(2) /(1) \Rightarrow 	an \phi=\left(\frac{x_0}{p_0}ight) m \omega$
$$\sin \phi=\frac{\mathrm{x}_0 \mathrm{~m} \omega}{\sqrt{\left(\mathrm{~m} \omega \mathrm{x}_0ight)^2+\mathrm{p}_0^2}}$$
From (1), $A=\frac{x_0}{\sin \phi}=\frac{\sqrt{\left(m \omega x_0ight)^2+p_0^2}}{m \omega}$
This means we can explain assertion with the given reason.

Answer

(A) is true but (R) is false

Answer

Both (A) and (R) are true but (R) is NOT the correct explanation of (A)

Answer

(A) is false but (R) is true

Answer

Both (A) and (R) are true and (R) is the correct explanation of (A)

Oscillations

Choose the correct answer: