Simple Harmonic Motion | JEE Main 2025

JEE MAIN 2025

28-01-2025 SHIFT-2

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 $x_{0}$ 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 $x_{0}$ and $p_{0}$ In the light of the above statements, choose the correct answer from the options given below:

Select the correct option:

(A) is false but (R) is true

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

(A) is true but (R) is false

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

✓ Correct! Well done.

✗ Incorrect. Try again or view the solution.

Solution

$\mathrm{x}=\mathrm{A} \sin (\omega \mathrm{t}+\phi) ; \mathrm{x}_{0}=\mathrm{A} \sin \phi$
$\mathrm{p}=\mathrm{mA} \omega \cos (\omega \mathrm{t}+\phi) \mathrm{p} ; p_{0}=m A \omega \cos \phi$
(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.

Question Tags

JEE Main

Physics

Medium

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