In a simple harmonic motion, when the displacement is one-half the amplitude, what fraction of the total energy is kinetic ?
(a) Zero (b) `(1)/(4)` (c) `(1)/(2)` (d) `(3)/(4)`

Options

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

Answer

To solve the problem, we need to determine the fraction of kinetic energy when the displacement is one-half of the amplitude in simple harmonic motion (SHM). Let's break it down step by step:

### Step 1: Define the Variables
Let:
- Amplitude (A) = A
- Displacement (x) = A/2 (since the displacement is one-half the amplitude)

### Step 2: Write the Expression for Kinetic Energy
In SHM, the kinetic energy (K.E.) can be expressed as:
\[ K.E. = \frac{1}{2} m \omega^2 (A^2 - x^2) \]
where:
- \( m \) is the mass,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude,
- \( x \) is the displacement.

### Step 3: Substitute the Displacement
Substituting \( x = \frac{A}{2} \) into the kinetic energy formula:
\[ K.E. = \frac{1}{2} m \omega^2 \left( A^2 - \left(\frac{A}{2}\right)^2 \right) \]

### Step 4: Simplify the Expression
Now simplify the expression:
\[ K.E. = \frac{1}{2} m \omega^2 \left( A^2 - \frac{A^2}{4} \right) \]
\[ K.E. = \frac{1}{2} m \omega^2 \left( \frac{4A^2}{4} - \frac{A^2}{4} \right) \]
\[ K.E. = \frac{1}{2} m \omega^2 \left( \frac{3A^2}{4} \right) \]
\[ K.E. = \frac{3}{8} m \omega^2 A^2 \]

### Step 5: Write the Expression for Total Energy
The total energy (E) in SHM is given by:
\[ E = \frac{1}{2} m \omega^2 A^2 \]

### Step 6: Find the Fraction of Kinetic Energy
Now, we need to find the fraction of kinetic energy to total energy:
\[ \text{Fraction} = \frac{K.E.}{E} \]
Substituting the values we found:
\[ \text{Fraction} = \frac{\frac{3}{8} m \omega^2 A^2}{\frac{1}{2} m \omega^2 A^2} \]

### Step 7: Simplify the Fraction
Canceling out the common terms:
\[ \text{Fraction} = \frac{3/8}{1/2} = \frac{3}{8} \times \frac{2}{1} = \frac{3}{4} \]

### Conclusion
Thus, the fraction of the total energy that is kinetic when the displacement is one-half the amplitude is:
\[ \boxed{\frac{3}{4}} \]