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

Options

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

Explanation

To solve the problem, we need to find the total 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 a given potential energy function \( U = 2r^3 \) J.

### Step-by-Step Solution:

1. **Identify the Potential Energy Function**:
The potential energy \( U \) is given by:
\[ U = 2r^3 \text{ J} \]

2. **Calculate the Force**:
The force \( F \) associated with the potential energy can be found using the relation:
\[ F = -\frac{dU}{dr} \]
First, we differentiate \( U \) with respect to \( r \):
\[ \frac{dU}{dr} = \frac{d}{dr}(2r^3) = 6r^2 \]
Therefore, the force is:
\[ F = -6r^2 \]

3. **Centripetal Force Requirement**:
For an object moving in a circular orbit, the centripetal force required is given by:
\[ F_c = \frac{mv^2}{r} \]
Setting the magnitude of the gravitational force equal to the centripetal force:
\[ \frac{mv^2}{r} = 6r^2 \]

4. **Express \( mv^2 \)**:
Rearranging the equation gives:
\[ mv^2 = 6r^3 \]

5. **Calculate Kinetic Energy**:
The kinetic energy \( K \) is given by:
\[ K = \frac{1}{2} mv^2 \]
Substituting \( mv^2 = 6r^3 \):
\[ K = \frac{1}{2} (6r^3) = 3r^3 \]

6. **Calculate Potential Energy**:
The potential energy at radius \( r \) is:
\[ U = 2r^3 \]

7. **Total Energy**:
The total energy \( E \) is the sum of kinetic and potential energy:
\[ E = K + U = 3r^3 + 2r^3 = 5r^3 \]

8. **Substituting \( r = 5 \) m**:
Now, substituting \( r = 5 \) m into the total energy equation:
\[ E = 5(5^3) = 5(125) = 625 \text{ J} \]

### Final Answer:
The total energy of the body is \( 625 \) Joules.

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