A particle is executing shm. What fraction of its energy is kinetic when the displacement is half the amplitude ?

Explanation

To solve the problem, we need to determine the fraction of kinetic energy when the displacement of a particle executing simple harmonic motion (SHM) is half of the amplitude.

### Step-by-Step Solution:

1. **Define the variables:**
- Let the amplitude of the SHM be \( A \).
- The displacement \( x \) is given as \( \frac{A}{2} \).

2. **Calculate the potential energy (PE) at displacement \( x = \frac{A}{2} \):**
- The formula for potential energy in SHM is given by:
\[ PE = \frac{1}{2} k x^2 \]
- Substituting \( x = \frac{A}{2} \):
\[ PE = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \frac{A^2}{4} = \frac{1}{8} k A^2 \]

3. **Determine the total energy (TE) in SHM:**
- The total energy in SHM is constant and is given by:
\[ TE = \frac{1}{2} k A^2 \]

4. **Relate potential energy to total energy:**
- From the above calculations, we have:
\[ PE = \frac{1}{8} k A^2 \]
- The total energy is:
\[ TE = \frac{1}{2} k A^2 \]

5. **Calculate the kinetic energy (KE):**
- According to the conservation of energy:
\[ TE = PE + KE \]
- Rearranging gives:
\[ KE = TE - PE \]
- Substituting the values we have:
\[ KE = \frac{1}{2} k A^2 - \frac{1}{8} k A^2 \]
- To combine these, convert \( \frac{1}{2} k A^2 \) into eighths:
\[ KE = \frac{4}{8} k A^2 - \frac{1}{8} k A^2 = \frac{3}{8} k A^2 \]

6. **Find the fraction of kinetic energy to total energy:**
- The fraction of kinetic energy to total energy is:
\[ \frac{KE}{TE} = \frac{\frac{3}{8} k A^2}{\frac{1}{2} k A^2} \]
- Simplifying this gives:
\[ \frac{KE}{TE} = \frac{3/8}{1/2} = \frac{3}{8} \times \frac{2}{1} = \frac{3}{4} \]

### Final Answer:
The fraction of the total energy that is kinetic when the displacement is half the amplitude is \( \frac{3}{4} \).