JEE Main 2025 Jan 28 Shift 2, Physics Q12: Simple Harmonic Motion And Oscillation

JEE Main2025Jan 28, Shift 2Physics

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 $ω$ .

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 :

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

Solution

Write the SHM solution as $x (t) = A sin (ω t + ϕ)$ . At $t = 0$ : $x_{0} = A sin ϕ$ and $p_{0} = m A ω cos ϕ$ .

These two equations can be inverted for the two unknowns $A$ and $ϕ$ :

$tan ϕ = \frac{mω x _{0}}{p _{0}}, A = x_{0}^{2} + \frac{p _{0}^{2}}{m ^{2} ω ^{2}}$ .

Once $A$ and $ϕ$ are known, $x (t)$ and $p (t) = m \overset{x}{˙} (t)$ are determined at every $t$ . So (A) is true, and (R) (which states this very fact) is true and serves as the correct explanation.

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JEE Main 2025 Jan 28 Shift 2, Physics Q12: Simple Harmonic Motion And Oscillation

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