Question

A body is doing SHM with amplitude A. When it is at \( x =+ \frac{A}{2}\) , find ratio of kinetic energy to potential energy.

• A. 1:1
• B. 3:1
• C. 2:1
• D. 4:1

Answer 1

The Correct Option is B

To solve the problem of finding the ratio of kinetic energy (KE) to potential energy (PE) when the body undergoing Simple Harmonic Motion (SHM) is at position \( x = \frac{A}{2} \), we need to understand the energy characteristics of SHM. Let's go through the steps:

1. In SHM, the total energy \(E\) of the system is constant and is given by: \( E = \frac{1}{2} k A^2 \), where \(k\) is the spring constant and \(A\) is the amplitude.
2. The potential energy at position \(x\) is: \( PE = \frac{1}{2} k x^2 \).
3. Substitute \(x = \frac{A}{2}\) into the potential energy formula: \( PE = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \frac{A^2}{4} = \frac{1}{8} k A^2 \).
4. The kinetic energy is then calculated by subtracting the potential energy from the total energy: \( KE = E - PE = \frac{1}{2} k A^2 - \frac{1}{8} k A^2 = \frac{4}{8} k A^2 - \frac{1}{8} k A^2 \) \( = \frac{3}{8} k A^2 \).
5. Thus, the ratio of kinetic energy to potential energy at \(x = \frac{A}{2}\) is: \[ \text{Ratio} = \frac{KE}{PE} = \frac{\frac{3}{8} k A^2}{\frac{1}{8} k A^2} = \frac{3}{1} \]

Therefore, the ratio of kinetic energy to potential energy when the body is at position \(x = \frac{A}{2}\) is 3:1. Hence, the correct answer is:

3:1