The maximum potential energy of a block executing simple harmonic motion is `25J.` A is amplitude of oscillation.At `A/2`,the kinetic energy of the block is

Options

A. `18.75J`
B. `37.5J`
C. `9.75J`
D. `12.5J`

Answer

To solve the problem, we need to determine the kinetic energy of a block executing simple harmonic motion (SHM) when it is at a position of \( \frac{A}{2} \), given that the maximum potential energy is \( 25 \, J \).

### Step-by-Step Solution:

1. **Understanding Maximum Potential Energy**:
The maximum potential energy \( U_{\text{max}} \) in SHM is given by the formula:
\[ U_{\text{max}} = \frac{1}{2} k A^2 \]
where \( k \) is the spring constant and \( A \) is the amplitude of the oscillation. We are given that \( U_{\text{max}} = 25 \, J \).

2. **Relating Potential Energy and Kinetic Energy**:
The total mechanical energy \( E \) in SHM is constant and is equal to the maximum potential energy:
\[ E = U_{\text{max}} = 25 \, J \]
At any position \( x \), the potential energy \( U \) and kinetic energy \( K \) are related by:
\[ E = K + U \]
Therefore, we can express the kinetic energy as:
\[ K = E - U \]

3. **Calculating Potential Energy at \( x = \frac{A}{2} \)**:
The potential energy \( U \) at position \( x = \frac{A}{2} \) can be calculated using the formula:
\[ U = \frac{1}{2} k x^2 \]
Substituting \( x = \frac{A}{2} \):
\[ U = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \frac{A^2}{4} = \frac{1}{8} k A^2 \]

4. **Finding \( k A^2 \)**:
From the maximum potential energy equation:
\[ U_{\text{max}} = \frac{1}{2} k A^2 = 25 \, J \]
We can solve for \( k A^2 \):
\[ k A^2 = 50 \, J \]

5. **Substituting \( k A^2 \) into the Potential Energy Equation**:
Now substituting \( k A^2 \) into the potential energy at \( x = \frac{A}{2} \):
\[ U = \frac{1}{8} k A^2 = \frac{1}{8} \times 50 \, J = 6.25 \, J \]

6. **Calculating Kinetic Energy at \( x = \frac{A}{2} \)**:
Now we can find the kinetic energy:
\[ K = E - U = 25 \, J - 6.25 \, J = 18.75 \, J \]

### Final Answer:
The kinetic energy of the block at \( x = \frac{A}{2} \) is \( 18.75 \, J \).