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:

• A. Both (A) and (R) are true and (R) is the correct explanation of (A)
• B. (A) is false but (R) is true
• C. Both (A) and (R) are true but (R) is NOT the correct explanation of (A)
• D. (A) is true but (R) is false

Hint:

In SHM problems, always express the amplitude and phase in terms of initial position and momentum to solve for them.

Answer 1

The Correct Option is A

We know that for simple harmonic motion, the position $x(t)$ and momentum $p(t)$ can be written as: $$x(t) = A \sin(\omega t + \phi)$$ $$p(t) = mA\omega \cos(\omega t + \phi)$$ From these, the amplitude $A$ and phase $\phi$ can be derived using initial conditions $x_0$ and $p_0$. Hence, (A) is true, and (R) provides the correct explanation for (A).