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 expressed as: \[ E_{\text{total}} = \frac{1}{2} k A^2, \] where \( k \) represents the spring constant and \( A \) denotes the amplitude of oscillation. Step 1: Potential and Kinetic Energy Expressions The potential energy (PE) at a displacement \( x \) is: \[ PE = \frac{1}{2} k x^2. \] The kinetic energy (KE) is the difference between the total energy and the potential energy: \[ KE = E_{\text{total}} - PE = \frac{1}{2} k A^2 - \frac{1}{2} k x^2. \] Step 2: Energy Calculations at Half Amplitude At a displacement \( x = \frac{A}{2} \): \[ PE = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \frac{A^2}{4} = \frac{1}{8} k A^2. \] And the kinetic energy is: \[ KE = \frac{1}{2} k A^2 - \frac{1}{8} k A^2 = \frac{4}{8} k A^2 - \frac{1}{8} k A^2 = \frac{3}{8} k A^2. \] Step 3: Ratio of Potential to Kinetic Energy The ratio of potential energy to kinetic energy is: \[ \frac{PE}{KE} = \frac{\frac{1}{8} k A^2}{\frac{3}{8} k A^2} = \frac{1}{3}. \] Final Answer: \[ \boxed{1:3} \]