Question

A particle is executing Simple Harmonic Motion (SHM). The ratio of potential energy and kinetic energy of the particle when its displacement is half of its amplitude will be:

• A. \( 1:1 \)
• B. \( 2:1 \)
• C. \( 1:4 \)
• D. \( 1:3 \)

Hint:

In SHM, potential energy is maximum at extreme displacement, while kinetic energy peaks at the equilibrium position. At intermediate displacements, use \( PE + KE = E_{\text{total}} \) to analyze energy distribution.

Answer 1

The Correct Option is D

The total energy in simple harmonic motion (SHM) is given by: \[ E_{\text{total}} = \frac{1}{2} k A^2, \] where: - \( k \) is the spring constant, - \( A \) is the amplitude of oscillation. Step 1: Expression for Potential and Kinetic Energy The potential energy at displacement \( x \) is: \[ PE = \frac{1}{2} k x^2. \] The kinetic energy is the remainder of the total energy: \[ KE = E_{\text{total}} - PE = \frac{1}{2} k A^2 - \frac{1}{2} k x^2. \] Step 2: Displacement at Half the Amplitude When \( x = \frac{A}{2} \): \[ PE = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \frac{A^2}{4} = \frac{1}{8} k A^2. \] \[ KE = \frac{1}{2} k A^2 - \frac{1}{8} k A^2 = \frac{4}{8} k A^2 - \frac{1}{8} k A^2 = \frac{3}{8} k A^2. \] Step 3: Ratio of Potential Energy to Kinetic Energy \[ \frac{PE}{KE} = \frac{\frac{1}{8} k A^2}{\frac{3}{8} k A^2} = \frac{1}{3}. \] Final Answer: \[ \boxed{1:3} \]