The total mechanical energy of a particle performing S.H.M. is 150 J with amplitude 1 m and force constant 200 N/m. Then minimum P.E. is,

Options

A. 50 J
B. 100 J
C. 150 J
D. zero

Explanation

To solve the problem, we need to find the minimum potential energy (P.E.) of a particle performing simple harmonic motion (S.H.M.) given the total mechanical energy, amplitude, and force constant. Let's break it down step by step.

### Step-by-Step Solution:

1. **Understand the Total Mechanical Energy in S.H.M.**:
The total mechanical energy (E) in S.H.M. is given by the formula:
\[ E = \frac{1}{2} k A^2 \]
where \( k \) is the force constant and \( A \) is the amplitude.

2. **Substitute the Given Values**:
We are given:
- Total mechanical energy \( E = 150 \, \text{J} \)
- Amplitude \( A = 1 \, \text{m} \)
- Force constant \( k = 200 \, \text{N/m} \)

Now, substituting the amplitude into the energy formula:
\[ E = \frac{1}{2} \times 200 \times (1)^2 = \frac{1}{2} \times 200 \times 1 = 100 \, \text{J} \]

3. **Analyze the Situation**:
We have calculated the maximum potential energy (P.E.) to be 100 J, but the total mechanical energy is given as 150 J. This indicates that the system is not in a typical S.H.M. scenario where maximum P.E. equals total mechanical energy.

4. **Find the Minimum Potential Energy**:
In S.H.M., the potential energy varies between 0 (at the mean position) and maximum (at the amplitude). The relationship between total mechanical energy, kinetic energy (K.E.), and potential energy (P.E.) is:
\[ E = K.E. + P.E. \]
To find the minimum potential energy, we can rearrange this equation:
\[ P.E. = E - K.E. \]
Since the minimum potential energy occurs when the kinetic energy is maximum, we can set \( K.E. \) to its maximum value when the total mechanical energy is at its maximum.

5. **Calculate Minimum P.E.**:
Given that the total mechanical energy is 150 J, we can find the minimum potential energy:
\[ P.E._{\text{min}} = E - K.E._{\text{max}} = 150 \, \text{J} - 100 \, \text{J} = 50 \, \text{J} \]

### Final Answer:
The minimum potential energy is:
\[ \text{Minimum P.E.} = 50 \, \text{J} \]