Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A): Knowing initial position x₀ and initial momentum p₀ 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 x₀ and p₀. In the light of the above statements, choose the correct answer from the options given below: (a) (A) is true but (R) is false (b) Both (A) and (R) are true but (R) is NOT the correct explanation of (A) (c) (A) is false but (R) is true (d) Both (A) and (R) are true and (R) is the correct explanation of (A)

Answer: (d). SHM: x = Asin (ω t + φ), p = mω Acos (ω t + φ). At t = 0: x₀ = Asin φ, p₀ = mω Acos φ, so A = √(x₀² + (p₀²)/(m²ω²)) and tan φ = (mω x₀)/(p₀). Once A and φ are known, x and p are known at every t. So R is true and explains A. So the answer is (d).

Given below are two statements. One is labelled as Assertion (A) and… | JEE Main 2025 (28 Jan, Shift 2)

Q16JEE Main 2025 · 28 JAN · SHIFT 2Easy

Periodic Motion & SHM Basics

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:

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Answer: (d)

SHM: $x = A\sin(\omega t + \phi)$ , $p = m\omega A\cos(\omega t + \phi)$ .
At $t = 0$ : $x_0 = A\sin\phi$ , $p_0 = m\omega A\cos\phi$ , so $A = \sqrt{x_0^2 + \frac{p_0^2}{m^2\omega^2}}$ and $\tan\phi = \frac{m\omega x_0}{p_0}$ .
Once $A$ and $\phi$ are known, $x$ and $p$ are known at every $t$ . So R is true and explains A.
So the answer is (d).

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