During motion of a planet from perihelion to aphelion the work done by gravitational force of sun on it is

Options

A. Zero
B. Negative
C. Positive
D. May be positive or negative

Answer

To solve the problem of the work done by the gravitational force of the Sun on a planet as it moves from perihelion to aphelion, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The planet moves in an elliptical orbit around the Sun. The closest point to the Sun is called perihelion, and the farthest point is called aphelion.

2. **Identifying the Forces**:
- The gravitational force exerted by the Sun on the planet is a central force. This means it acts along the line connecting the center of the Sun and the planet.

3. **Work Done by Gravitational Force**:
- The work done by a force is defined as the change in kinetic energy of the object. According to the work-energy theorem:
\[ W = \Delta KE = KE_{final} - KE_{initial} \]

4. **Kinetic Energy at Perihelion and Aphelion**:
- Let \( v_p \) be the velocity of the planet at perihelion and \( v_a \) be the velocity at aphelion. The kinetic energies can be expressed as:
\[ KE_{perihelion} = \frac{1}{2} m v_p^2 \]
\[ KE_{aphelion} = \frac{1}{2} m v_a^2 \]

5. **Conservation of Angular Momentum**:
- The angular momentum of the planet is conserved. Therefore, we have:
\[ m v_p r_p = m v_a r_a \]
where \( r_p \) and \( r_a \) are the distances from the Sun to the planet at perihelion and aphelion, respectively. This implies:
\[ v_p r_p = v_a r_a \]

6. **Relationship Between Velocities and Distances**:
- Since \( r_p < r_a \) (the planet is closer to the Sun at perihelion), it follows that:
\[ v_p > v_a \]
- This means the velocity at perihelion is greater than the velocity at aphelion.

7. **Calculating Work Done**:
- Now we can calculate the work done:
\[ W = KE_{aphelion} - KE_{perihelion} = \frac{1}{2} m v_a^2 - \frac{1}{2} m v_p^2 \]
- Since \( v_p > v_a \), it follows that:
\[ W < 0 \]
- Therefore, the work done by the gravitational force of the Sun on the planet as it moves from perihelion to aphelion is negative.

### Conclusion:
The work done by the gravitational force of the Sun on the planet during its motion from perihelion to aphelion is negative.

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