Question

A body performs linear simple harmonic motion of amplitude $A$. At what displacement from the mean position, the potential energy of the body is one fourth of its total energy?

• A. $\frac{A}{3}$
• B. $\frac{A}{2}$
• C. $\frac{3A}{4}$
• D. $\frac{A}{4}$

Hint:

For simple harmonic motion, when potential energy is $\frac{1}{4}$ of total energy, the displacement is half of the amplitude.

Answer 1

The Correct Option is B

Step 1: Energy in SHM.
Total energy in simple harmonic motion is constant and given by:
\[ E = \frac{1}{2}kA^2 \]
Step 2: Potential energy in SHM.
Potential energy is given by:
\[ U = \frac{1}{2}k x^2 \]
Step 3: Condition for potential energy.
At the required displacement, potential energy is one fourth of total energy:
\[ \frac{U}{E} = \frac{1}{4} \]
Step 4: Solving for displacement.
\[ \frac{x^2}{A^2} = \frac{1}{4} \] \[ x = \frac{A}{2} \]
Step 5: Conclusion.
The displacement is $\frac{A}{2}$.