Question

A particle is executing simple harmonic motion with a time period of 3 s. At a position where the displacement of the particle is 60% of its amplitude, the ratio of the kinetic and potential energies of the particle is:

• A. 5 : 3
• B. 16 : 9
• C. 4 : 3
• D. 25 : 9

Hint:

N/A

Answer 1

The Correct Option is B

We are given a particle executing simple harmonic motion (SHM) with a time period \( T = 3 \, \text{s} \). The displacement of the particle is 60% of its amplitude. We are asked to find the ratio of kinetic and potential energies at this position. 
Step 1: Relationship between displacement, velocity, and energy For SHM, the displacement of the particle is given by: \[ x = A \sin(\omega t), \] where \( A \) is the amplitude, and \( \omega \) is the angular frequency. The angular frequency \( \omega \) is related to the time period by: \[ \omega = \frac{2\pi}{T}. \] The total mechanical energy in SHM is given by: \[ E_{\text{total}} = \frac{1}{2} m \omega^2 A^2. \] This energy is constant, and it is the sum of the kinetic energy (\( K \)) and potential energy (\( U \)): \[ E_{\text{total}} = K + U. \] 
Step 2: Kinetic and potential energies At any point during the motion, the kinetic energy is: \[ K = \frac{1}{2} m \omega^2 (A^2 - x^2), \] and the potential energy is: \[ U = \frac{1}{2} m \omega^2 x^2. \] Now, we are given that the displacement \( x = 0.6A \), so we can substitute this into the equations for \( K \) and \( U \). The kinetic energy at \( x = 0.6A \) is: \[ K = \frac{1}{2} m \omega^2 \left( A^2 - (0.6A)^2 \right) = \frac{1}{2} m \omega^2 \left( A^2 - 0.36A^2 \right) = \frac{1}{2} m \omega^2 \times 0.64A^2. \] The potential energy at \( x = 0.6A \) is: \[ U = \frac{1}{2} m \omega^2 (0.6A)^2 = \frac{1}{2} m \omega^2 \times 0.36A^2. \] 
Step 3: Ratio of kinetic to potential energy Now, we can find the ratio of \( K \) to \( U \): \[ \frac{K}{U} = \frac{\frac{1}{2} m \omega^2 \times 0.64A^2}{\frac{1}{2} m \omega^2 \times 0.36A^2} = \frac{0.64}{0.36} = \frac{16}{9}. \] 
Thus, the ratio of kinetic energy to potential energy is \( 16 : 9 \). Therefore, the correct answer is option (2).