The potential energy of a particle executing S H M is 25 J. when its displacement is half of amplitude. The total energy of the particle is

Options

A. 250 J
B. 180 J
C. 100 J
D. 25 J

Answer

To solve the problem, we need to find the total energy of a particle executing Simple Harmonic Motion (SHM) given that its potential energy is 25 J when its displacement is half of the amplitude.

### Step-by-Step Solution:

1. **Understand the relationship between potential energy and displacement in SHM**:
The potential energy (PE) of a particle in SHM is given by the formula:
\[ PE = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

2. **Identify the amplitude**:
Let the amplitude of the motion be \( A \). According to the problem, the displacement \( x \) is half of the amplitude:
\[ x = \frac{A}{2} \]

3. **Substitute the displacement into the potential energy formula**:
Substituting \( x = \frac{A}{2} \) into the potential energy formula:
\[ PE = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \cdot \frac{A^2}{4} = \frac{k A^2}{8} \]

4. **Set the potential energy equal to the given value**:
We know from the problem that the potential energy is 25 J:
\[ \frac{k A^2}{8} = 25 \]
Multiplying both sides by 8 gives:
\[ k A^2 = 200 \]

5. **Total energy in SHM**:
The total energy (TE) in SHM is given by the formula:
\[ TE = \frac{1}{2} k A^2 \]
From the previous step, we have \( k A^2 = 200 \). Therefore, we can substitute this into the total energy formula:
\[ TE = \frac{1}{2} \cdot 200 = 100 \text{ J} \]

6. **Final answer**:
The total energy of the particle is:
\[ \text{Total Energy} = 100 \text{ J} \]