In a simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the total energy is kinetic ?

Options

A. `(1)/(2)`
B. `(3)/(4)`
C. zero
D. `(1)/(4)`

Answer

To solve the problem step by step, we will use the concepts of simple harmonic motion (SHM) and the formulas for kinetic energy and total energy.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to find the fraction of kinetic energy (KE) when the displacement (x) is half of the amplitude (A) in a simple harmonic motion.

2. **Setting Up the Displacement**:
\[ x = \frac{1}{2} A \]

3. **Kinetic Energy Formula**: The kinetic energy (KE) in SHM is given by:
\[ KE = \frac{1}{2} mv^2 \]
where \( v \) is the velocity of the mass.

4. **Velocity in SHM**: The velocity \( v \) can be expressed in terms of angular frequency \( \omega \) and displacement \( x \):
\[ v = \omega \sqrt{A^2 - x^2} \]

5. **Substituting Displacement**: Since \( x = \frac{1}{2} A \), we can substitute this into the velocity equation:
\[ v = \omega \sqrt{A^2 - \left(\frac{1}{2} A\right)^2} \]
Simplifying this gives:
\[ v = \omega \sqrt{A^2 - \frac{1}{4} A^2} = \omega \sqrt{\frac{3}{4} A^2} = \omega \frac{\sqrt{3}}{2} A \]

6. **Calculating Kinetic Energy**: Now substituting \( v \) back into the kinetic energy formula:
\[ KE = \frac{1}{2} m \left(\omega \frac{\sqrt{3}}{2} A\right)^2 = \frac{1}{2} m \left(\frac{3}{4} \omega^2 A^2\right) = \frac{3}{8} m \omega^2 A^2 \]

7. **Total Energy in SHM**: The total energy (TE) in SHM is given by:
\[ TE = \frac{1}{2} m \omega^2 A^2 \]

8. **Finding the Fraction of Kinetic Energy to Total Energy**:
\[ \text{Fraction} = \frac{KE}{TE} = \frac{\frac{3}{8} m \omega^2 A^2}{\frac{1}{2} m \omega^2 A^2} \]
Simplifying this gives:
\[ \text{Fraction} = \frac{3/8}{1/2} = \frac{3}{8} \cdot \frac{2}{1} = \frac{3}{4} \]

9. **Conclusion**: Therefore, when the displacement is one-half the amplitude, the fraction of the total energy that is kinetic energy is:
\[ \frac{3}{4} \]

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