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).

Answer 2

Step 1: Understanding the Assertion (A) and 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 (SHM) with a given angular frequency \( \omega \).
Reason (R): The amplitude and phase can be expressed in terms of \( x_0 \) and \( p_0 \).

Step 2: Mathematical form of SHM.
The displacement in SHM is given by:
\[ x(t) = A \cos(\omega t + \phi) \] and the momentum is given by:
\[ p(t) = m \frac{dx}{dt} = -m\omega A \sin(\omega t + \phi). \]
At \( t = 0 \):
\[ x_0 = A \cos \phi, \quad p_0 = -m\omega A \sin \phi. \]

Step 3: Express amplitude and phase in terms of \( x_0 \) and \( p_0 \).
Squaring and adding these two equations:
\[ x_0^2 + \left(\frac{p_0}{m\omega}\right)^2 = A^2. \] Hence, \[ A = \sqrt{x_0^2 + \left(\frac{p_0}{m\omega}\right)^2}. \] Dividing the two expressions: \[ \tan \phi = -\frac{p_0}{m\omega x_0}. \] Therefore, both \( A \) and \( \phi \) can be fully determined if \( x_0 \) and \( p_0 \) are known.

Step 4: Determining motion at any time.
Once \( A \) and \( \phi \) are known, the complete motion is determined since:
\[ x(t) = A \cos(\omega t + \phi), \quad p(t) = -m\omega A \sin(\omega t + \phi). \] Thus, knowing \( x_0 \) and \( p_0 \) is sufficient to find position and momentum at any time \( t \).

Step 5: Conclusion.
Both (A) and (R) are true. The reason correctly explains why knowing \( x_0 \) and \( p_0 \) is sufficient — because they determine amplitude \( A \) and phase \( \phi \), which fully describe SHM.

Final Answer:
Both (A) and (R) are true and (R) is the correct explanation of (A).
\[ \boxed{\text{Both (A) and (R) are true and (R) is the correct explanation of (A).}} \]