The potential energy of a particle execuring S.H.M. is 5 J, when its displacement is half of amplitude. The total energy of the particle be

Options

A. 18 J
B. 10 J
C. 12J
D. 2.5J

Explanation

To find the total energy of a particle executing Simple Harmonic Motion (SHM) given that its potential energy is 5 J when its displacement is half of the amplitude, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship between potential energy and total energy in SHM**:
The total energy (E) of a particle in SHM is given by the formula:
\[ E = \frac{1}{2} k A^2 \]
where \(k\) is the spring constant and \(A\) is the amplitude.

2. **Write the expression for potential energy**:
The potential energy (PE) at a displacement \(x\) in SHM is given by:
\[ PE = \frac{1}{2} k x^2 \]

3. **Substitute the given displacement**:
We are given that the displacement \(x\) is half of the amplitude, so:
\[ x = \frac{A}{2} \]

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

5. **Set the potential energy equal to the given value**:
We know from the problem that this potential energy is 5 J:
\[ \frac{1}{8} k A^2 = 5 \]

6. **Relate this to total energy**:
We know that the total energy \(E\) can also be expressed in terms of \(k\) and \(A\):
\[ E = \frac{1}{2} k A^2 \]
To express \(E\) in terms of the known potential energy, we can relate the two:
\[ k A^2 = 40 \quad (\text{from } \frac{1}{8} k A^2 = 5 \Rightarrow k A^2 = 40) \]

7. **Substitute back to find total energy**:
Now substitute \(k A^2 = 40\) into the total energy formula:
\[ E = \frac{1}{2} (40) = 20 \text{ J} \]

### Final Answer:
The total energy of the particle is **20 Joules**.

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