Question

When the displacement of a particle executing simple harmonic motion is half its amplitude, the ratio of its kinetic energy to potential energy is:

• A. 1 : 3
• B. 2 : 1
• C. 3 : 1
• D. 1 : 2
• E. 2 : 3

Hint:

At $y = A/2$, $PE$ is $25%$ of total energy and $KE$ is $75%$ of total energy. Thus the ratio is $75:25 = 3:1$.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
In Simple Harmonic Motion (SHM), the total energy is conserved and is the sum of Kinetic Energy ($KE$) and Potential Energy ($PE$).
Step 2: Key Formula or Approach:
Potential Energy: $PE = \frac{1}{2} k y^2$
Kinetic Energy: $KE = \frac{1}{2} k (A^2 - y^2)$
where $A$ is amplitude and $y$ is displacement.
Step 3: Detailed Explanation:
Given $y = \frac{A}{2}$.
$PE = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \frac{A^2}{4} = \frac{1}{8} k A^2$
$KE = \frac{1}{2} k \left(A^2 - \left(\frac{A}{2}\right)^2\right) = \frac{1}{2} k \left(A^2 - \frac{A^2}{4}\right) = \frac{1}{2} k \left(\frac{3A^2}{4}\right) = \frac{3}{8} k A^2$
Calculating the ratio $\frac{KE}{PE}$: \[ \frac{KE}{PE} = \frac{\frac{3}{8} k A^2}{\frac{1}{8} k A^2} = \frac{3}{1} \] Step 4: Final Answer:
The ratio is 3 : 1.