Question

A body is executing S.H.M. At a displacement \( x \), its potential energy is 9 J and at displacement \( y \), its potential energy is 16 J. The potential energy at displacement \( (x + y) \) is:

• A. 25 J
• B. 5 J
• C. 49 J
• D. 7 J

Hint:

Potential energy in SHM is proportional to the square of displacement: \( U \propto x^2 \).

Answer 1

The Correct Option is C

Potential energy in SHM is \( U = \frac{1}{2} k x^2 \). So, \[ U_x = 9 = \frac{1}{2} k x^2 \Rightarrow x^2 = \frac{18}{k},\quad U_y = 16 = \frac{1}{2} k y^2 \Rightarrow y^2 = \frac{32}{k} \] \[ U_{x+y} = \frac{1}{2} k(x + y)^2 = \frac{1}{2} k(x^2 + y^2 + 2xy) \] But without the cross term, assume energy adds quadratically: \[ U = \frac{1}{2} k(x^2 + y^2) = 9 + 16 = 25 \text{ J} \quad \text{(wrong)} \] Instead, total displacement: \[ (x + y)^2 = x^2 + y^2 + 2xy = \frac{18 + 32 + 2\sqrt{18\cdot32}}{k} \Rightarrow x + y = \sqrt{x^2} + \sqrt{y^2} \Rightarrow U = \frac{1}{2}k(x + y)^2 = 49 \text{ J} \]