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 moving from perihelion to aphelion, we can follow these steps:

### Step 1: Understand the Definitions
- **Perihelion** is the point in the orbit of a planet where it is closest to the Sun.
- **Aphelion** is the point where the planet is farthest from the Sun.

### Step 2: Apply the Work-Energy Theorem
The work done by the gravitational force can be calculated using the work-energy theorem, which states that the work done by all forces acting on an object is equal to the change in its kinetic energy.

\[ W = K_f - K_i \]

Where:
- \( W \) is the work done,
- \( K_f \) is the final kinetic energy at aphelion,
- \( K_i \) is the initial kinetic energy at perihelion.

### Step 3: Express Kinetic Energy
The kinetic energy \( K \) of an object is given by:

\[ K = \frac{1}{2} mv^2 \]

Where \( m \) is the mass of the planet and \( v \) is its velocity.

### Step 4: Determine Velocities at Perihelion and Aphelion
At perihelion (point closest to the Sun), the planet has a higher velocity (\( V_p \)), and at aphelion (point farthest from the Sun), it has a lower velocity (\( V_a \)). This is due to the conservation of angular momentum, which states:

\[ L = mvr \quad \text{(constant)} \]

Since the radius at perihelion (\( r_p \)) is smaller than the radius at aphelion (\( r_a \)), the velocity at perihelion must be greater than that at aphelion:

\[ V_p > V_a \]

### Step 5: Calculate the Change in Kinetic Energy
Now we can express the change in kinetic energy:

\[ K_i = \frac{1}{2} m V_p^2 \]
\[ K_f = \frac{1}{2} m V_a^2 \]

Substituting these into the work-energy theorem:

\[ W = \frac{1}{2} m V_a^2 - \frac{1}{2} m V_p^2 \]

### Step 6: Analyze the Sign of Work Done
Since \( V_p > V_a \), it follows that:

\[ V_p^2 > V_a^2 \]

Thus, the work done becomes:

\[ W = \frac{1}{2} m V_a^2 - \frac{1}{2} m V_p^2 < 0 \]

This indicates that 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.

---