Question

A body performing simple harmonic motion has potential energy \(P_1\) at displacement \(x_1\). Its potential energy is \(P_2\) at displacement \(x_2\). The potential energy \(P\) at displacement \((x_1 + x_2)\) is

• A. \(P_1 + P_2\)
• B. \( \sqrt{P_1 P_2} \)
• C. \( \sqrt{P_1^2 + P_2^2} \)
• D. \( P_1 + P_2 + 2\sqrt{P_1 P_2} \)

Hint:

In SHM, potential energy varies as square of displacement.

Answer 1

The Correct Option is D

Step 1: Potential energy in SHM.
Potential energy is proportional to the square of displacement: \[ P = \frac{1}{2}kx^2. \]
Step 2: Expressing energies.
\[ P_1 = \frac{1}{2}k x_1^2, \quad P_2 = \frac{1}{2}k x_2^2. \]
Step 3: Energy at displacement \((x_1 + x_2)\).
\[ P = \frac{1}{2}k(x_1 + x_2)^2 = \frac{1}{2}k(x_1^2 + x_2^2 + 2x_1x_2). \] \[ P = P_1 + P_2 + 2\sqrt{P_1P_2}. \]
Step 4: Conclusion.
The correct expression is \( P_1 + P_2 + 2\sqrt{P_1 P_2} \).