Question

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R). 
Assertion (A): Knowing the 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).