Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): Knowing initial position and initial momentum is enough to determine the position and momentum at any ime t for a simple harmonic m

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Solution

Q.
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: [2025]

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

2 (A) is false but (R) is true.

3 (A) is true but (R) is false.

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

Ans.
(4)

$x = A \sin (ω t + ϕ)$

$x_{0} = A \sin ϕ$ ... (i)

$p = m A ω \cos (ω t + ϕ)$

$p_{0} = m A ω \cos ϕ$ ... (ii)

(ii)/(i)

$\Rightarrow \tan ϕ = (\frac{x_{0}}{p_{0}}) m ω$

$\sin ϕ = \frac{x_{0} m ω}{\sqrt{{(m ω x_{0})}^{2} + p_{0}^{2}}}$

From (i),

$A = \frac{x_{0}}{\sin ϕ} = \frac{\sqrt{{(m ω x_{0})}^{2} + p_{0}^{2}}}{m ω}$

Hence both position and linear momentum of a particle can be expressed as a function of time if we know initial momentum and position.

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