Question

A body is executing simple harmonic motion. At a displacement \( x \), its potential energy is \( E_1 \), and at a displacement \( y \), its potential energy is \( E_2 \). The potential energy \( E \) at a displacement \( (x + y) \) is:

• A. \(\sqrt{E} = \sqrt{E_1} - \sqrt{E_2} \)
• B. \(\sqrt{E} = \sqrt{E_1} + \sqrt{E_2} \)
• C. \(E = E_1 - E_2\)
• D. \(E = E_1 + E_2\)

Hint:

In simple harmonic motion, the potential energy follows a quadratic relation with displacement, leading to the sum of square roots property.

Answer 1

The Correct Option is B

Step 1: Understanding Potential Energy in SHM The potential energy in SHM is given by: \[ E = \frac{1}{2} k x^2 \] where \( k \) is the force constant. Step 2: Applying to Given Displacements For two different displacements: \[ E_1 = \frac{1}{2} k x^2, \quad E_2 = \frac{1}{2} k y^2 \] For total displacement \( (x+y) \): \[ E = \frac{1}{2} k (x+y)^2 \] Step 3: Using the Energy Sum Property Since energy follows a quadratic relationship: \[ \sqrt{E} = \sqrt{E_1} + \sqrt{E_2} \] Thus, the correct answer is option (2).