Passage I) In simple harmonic motion force acting on a particle is given as `F=-4x`, total mechanical energy of the particle is 10 J and amplitude of oscillations is 2m, At time t=0 acceleration of the particle is `-16m/s^(2)`. Mass of the particle is 0.5 kg.
Potential energy of the particle at mean position is

Options

A. 10 J
B. 8 J
C. 6 J
D. 2 J

Answer

To find the potential energy of the particle at the mean position in simple harmonic motion (SHM), we can follow these steps:

### Step 1: Understand the relationship between total mechanical energy, potential energy, and kinetic energy.
In SHM, the total mechanical energy (E) is the sum of kinetic energy (K) and potential energy (U). At the mean position, the potential energy is at its minimum (U = 0), and the kinetic energy is at its maximum. The relationship can be expressed as:
\[ E = K + U \]

### Step 2: Identify the given values.
From the problem, we have:
- Total mechanical energy (E) = 10 J
- Mass of the particle (m) = 0.5 kg
- Amplitude (A) = 2 m
- Force acting on the particle (F) = -4x
- Acceleration at t = 0 = -16 m/s²

### Step 3: Calculate the spring constant (k).
From the force equation \( F = -kx \), we can identify that \( k = 4 \) N/m (since \( F = -4x \)).

### Step 4: Calculate the maximum velocity (v_max).
The maximum velocity in SHM can be expressed as:
\[ v_{\text{max}} = \omega A \]
where \( \omega = \sqrt{\frac{k}{m}} \).

First, calculate \( \omega \):
\[ \omega = \sqrt{\frac{4}{0.5}} = \sqrt{8} = 2\sqrt{2} \, \text{rad/s} \]

Now, calculate \( v_{\text{max}} \):
\[ v_{\text{max}} = \omega A = (2\sqrt{2}) \cdot 2 = 4\sqrt{2} \, \text{m/s} \]

### Step 5: Calculate the maximum kinetic energy (K).
The maximum kinetic energy can be calculated using the formula:
\[ K = \frac{1}{2} m v_{\text{max}}^2 \]
Substituting the values:
\[ K = \frac{1}{2} \cdot 0.5 \cdot (4\sqrt{2})^2 \]
\[ K = 0.25 \cdot 32 = 8 \, \text{J} \]

### Step 6: Calculate the potential energy (U) at the mean position.
Using the total mechanical energy equation:
\[ E = K + U \]
We can rearrange this to find U:
\[ U = E - K \]
Substituting the known values:
\[ U = 10 \, \text{J} - 8 \, \text{J} = 2 \, \text{J} \]

### Conclusion
The potential energy of the particle at the mean position is \( 2 \, \text{J} \).