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 an p₀.
In the light of the above statements, choose the correct answer from the options given below :

Solution

Physics sir · 4 steps

With ω fixed, SHM has just two unknowns, A and φ, and x₀ and p₀ pin both down.

1. Write the motion. With ω given, SHM is x = A sin(ω t + φ) and p = mω A cos(ω t + φ). Only the amplitude A and the phase φ are still unknown.
2. Use the start. At t = 0, x₀ = A sin φ and p₀ = mω A cos φ. That gives two equations for two unknowns.

   A = √(x₀² + (p₀/mω)²), tan φ = (mω x₀)/p₀

3. Check each statement. Reason (R) is true: A and φ are fixed by x₀ and p₀, since the particle's mass is known. Once A and φ are known, x(t) and p(t) are known at every t. So Assertion (A) is true because of (R).
4. So. Both (A) and (R) are true, and (R) is the correct explanation of (A). The answer is (d).

The idea to keep. SHM with a known ω has just two constants, amplitude and phase. Any two independent starting values, like position and momentum, fix them and so fix the whole motion.

Answer from the NTA final answer key. Solution by PrepWiser.