A body of mass `2 kg` is moving under the influence of a central force whose potential energy is given by `U = 2r^(3) J`. If the body is moving in a circular orbit of `5 m`, its energy will be a) 625 J b) 250 J c) 500 J d) 125 J

Options

A. `625 J`
B. `250 J`
C. `500 J`
D. `125 J`

Explanation

To solve the problem, we need to find the total mechanical energy of a body of mass 2 kg moving in a circular orbit of radius 5 m under the influence of a central force with the given potential energy \( U = 2r^3 \) J.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Mass of the body, \( m = 2 \, \text{kg} \)
- Radius of the circular orbit, \( r = 5 \, \text{m} \)
- Potential energy, \( U = 2r^3 \)

2. **Calculate the Potential Energy**:
\[ U = 2r^3 = 2(5^3) = 2(125) = 250 \, \text{J} \]

3. **Determine the Force**:
The force associated with the potential energy can be calculated using the formula:
\[ F = -\frac{dU}{dr} \]
First, we differentiate \( U \):
\[ U = 2r^3 \implies \frac{dU}{dr} = 6r^2 \]
Therefore, the force is:
\[ F = -6r^2 \]

4. **Calculate the Centripetal Force**:
Since the body is moving in a circular orbit, the centripetal force \( F_c \) is given by:
\[ F_c = \frac{mv^2}{r} \]
Setting the centripetal force equal to the force derived from the potential energy:
\[ \frac{mv^2}{r} = 6r^2 \]

5. **Solve for \( v^2 \)**:
Rearranging the equation gives:
\[ mv^2 = 6r^3 \implies v^2 = \frac{6r^3}{m} \]
Substituting \( m = 2 \, \text{kg} \) and \( r = 5 \, \text{m} \):
\[ v^2 = \frac{6(5^3)}{2} = \frac{6(125)}{2} = 375 \, \text{m}^2/\text{s}^2 \]

6. **Calculate the Kinetic Energy**:
The kinetic energy \( K \) is given by:
\[ K = \frac{1}{2} mv^2 = \frac{1}{2} (2)(375) = 375 \, \text{J} \]

7. **Calculate Total Mechanical Energy**:
The total mechanical energy \( E \) is the sum of kinetic energy and potential energy:
\[ E = K + U = 375 \, \text{J} + 250 \, \text{J} = 625 \, \text{J} \]

8. **Final Answer**:
The total energy of the body is \( 625 \, \text{J} \), which corresponds to option (a).