The total energy of a simple pendulum is x. When the displacement is half of amplitude,its KE will be

Options

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

Answer

To solve the problem, we need to determine the kinetic energy (KE) of a simple pendulum when its displacement is half of the amplitude. The total energy (TE) of the pendulum is given as \( x \).

### Step-by-Step Solution:

1. **Understanding the Energy of the Pendulum**:
The total mechanical energy \( E \) of a simple pendulum is constant and is given by the sum of its kinetic energy \( KE \) and potential energy \( PE \):
\[ E = KE + PE \]

2. **Define the Amplitude and Displacement**:
Let the amplitude of the pendulum be \( A \). According to the problem, the displacement \( d \) is half of the amplitude:
\[ d = \frac{A}{2} \]

3. **Calculate Potential Energy at Displacement**:
The potential energy \( PE \) at a displacement \( d \) can be calculated using the formula:
\[ PE = mgh \]
For small angles, we can approximate \( h \) (the height) using the formula:
\[ h = A - \sqrt{A^2 - d^2} \]
Substituting \( d = \frac{A}{2} \):
\[ h = A - \sqrt{A^2 - \left(\frac{A}{2}\right)^2} = A - \sqrt{A^2 - \frac{A^2}{4}} = A - \sqrt{\frac{3A^2}{4}} = A - \frac{\sqrt{3}}{2}A = A\left(1 - \frac{\sqrt{3}}{2}\right) \]

4. **Calculate Total Energy**:
The total energy \( E \) is given as \( x \). At maximum displacement (amplitude), the potential energy is maximum and equals the total energy:
\[ E = \frac{1}{2}kA^2 \]
where \( k \) is the spring constant equivalent for the pendulum.

5. **Calculate Kinetic Energy**:
Now, we can express the kinetic energy \( KE \) as:
\[ KE = E - PE \]
Substituting the values we have:
\[ KE = x - PE \]

6. **Finding the Kinetic Energy**:
We can express the potential energy at \( d = \frac{A}{2} \) in terms of \( x \) and substitute:
\[ PE = \frac{1}{2}k\left(\frac{A}{2}\right)^2 = \frac{1}{2}k\frac{A^2}{4} = \frac{1}{8}kA^2 \]
Since \( E = \frac{1}{2}kA^2 = x \), we can relate \( kA^2 \) to \( x \):
\[ PE = \frac{1}{8}\left(2x\right) = \frac{x}{4} \]
Thus,
\[ KE = x - \frac{x}{4} = \frac{3x}{4} \]

### Final Answer:
The kinetic energy when the displacement is half of the amplitude is:
\[ KE = \frac{3x}{4} \]