Different points in the earth are at slightly different distances from the sun and hence experience different forces due to gravitation.
For a rigid body, we know that if various forces act at various points in it, the resulant motion is as if net force acts on the CM (centre of mass) causing translation and a net torque at the CM causing rotation around an axis through the CM. For the earth-sun system (approximating the earth as a uniform density sphere)

Options

A. the torque is zero
B. the torque causes the earth to spin
C. the rigid body result is not applicable since the earth is not even approximately a rigid body
D. the torque causes the earth to move around the sun

Explanation

To solve the problem, we need to analyze the gravitational forces acting on different points of the Earth due to the Sun and how these forces relate to the motion of the Earth as a rigid body. Here’s a step-by-step solution:

### Step 1: Understand the Gravitational Force
Different points on the Earth are at varying distances from the Sun. This means that each point experiences a different gravitational force due to the Sun. The force acting on any point can be described by Newton's law of gravitation.

**Hint:** Remember that gravitational force decreases with the square of the distance between two masses.

### Step 2: Concept of Center of Mass
For a rigid body like the Earth, we can simplify our analysis by considering all the mass to be concentrated at a single point known as the center of mass (CM). The resultant motion of the Earth can be analyzed by considering the net force acting on the CM.

**Hint:** The center of mass is the point where the total mass of the body can be considered to act.

### Step 3: Resultant Force and Torque
When multiple forces act on a rigid body, the net force causes translational motion of the CM, and the net torque about the CM causes rotational motion. The torque (τ) can be calculated using the formula:
\[ \tau = R \times F \]
where \( R \) is the position vector from the CM to the point of force application, and \( F \) is the force vector.

**Hint:** Torque is a measure of the tendency of a force to rotate an object about an axis.

### Step 4: Analyze the Torque in the Earth-Sun System
In the case of the Earth and Sun, the gravitational force acts along the line connecting their centers. This means that the angle (θ) between the position vector (R) and the force vector (F) is 0 degrees. Therefore, the torque can be calculated as:
\[ \tau = R \cdot F \cdot \sin(θ) \]
Since \( \sin(0) = 0 \), the torque becomes:
\[ \tau = R \cdot F \cdot 0 = 0 \]

**Hint:** When the angle between the force and the position vector is 0 or 180 degrees, the torque is zero.

### Step 5: Conclusion on Options
1. **Torque is zero:** This is correct because we calculated that the torque acting on the Earth due to the Sun is zero.
2. **Torque causes the Earth to spin:** This is incorrect because if torque is zero, it cannot cause rotation.
3. **Rigid body result is not applicable:** This is incorrect in this context, as we are approximating the Earth as a uniform density sphere.
4. **Torque causes the Earth to move around the Sun:** This is incorrect; the Earth moves around the Sun due to gravitational attraction, not torque.

**Final Answer:** The correct option is that the torque is zero.