JEE MAIN - Physics (2025 - 28th January Evening Shift - No. 19)
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 :
Explanation
$$\begin{aligned} & \mathrm{x}=\mathrm{A} \sin (\omega \mathrm{t}+\phi) \\ & \mathrm{x}_0=\mathrm{A} \sin \phi \quad\text{.... (1)}\\ & \mathrm{p}=\mathrm{mA} \omega \cos (\omega \mathrm{t}+\phi) \\ & \mathrm{p}_0=\mathrm{mA} \omega \cos \phi \quad\text{.... (2)} \end{aligned}$$
$(2) /(1) \Rightarrow \tan \phi=\left(\frac{x_0}{p_0}\right) m \omega$
$$\sin \phi=\frac{\mathrm{x}_0 \mathrm{~m} \omega}{\sqrt{\left(\mathrm{~m} \omega \mathrm{x}_0\right)^2+\mathrm{p}_0^2}}$$
From (1), $A=\frac{x_0}{\sin \phi}=\frac{\sqrt{\left(m \omega x_0\right)^2+p_0^2}}{m \omega}$
This means we can explain assertion with the given reason.
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