An object is taken from a point `P` to another point `Q` in a gravitational field:

Options

A. assuming the earth to be spherical, if both `P` and `Q` lie on the earth's surface, the work done is zero
B. if `P` is on the earth's surface and `Q` above it, the work done is minimum when it is taken along the straight line `PQ`
C. the work done depends only on the position of `P` and `Q` and is independent of the path along which the particle is taken
D. there is no work done if the object is taken from `P` to `Q` and then brought back to `P` along any path

Explanation

To solve the problem, we need to analyze the work done when an object is moved in a gravitational field from point P to point Q. We will evaluate the given options one by one.

### Step-by-Step Solution:

1. **Understanding Work Done in a Gravitational Field**:
Work done (W) in a gravitational field is defined as the force applied (F) multiplied by the displacement (d) in the direction of the force. Mathematically, W = F * d * cos(θ), where θ is the angle between the force and the displacement.

2. **Option 1: Both P and Q lie on Earth's Surface**:
If both points P and Q are on the Earth's surface, they are at the same height from the center of the Earth. Since gravitational potential energy depends on height, and there is no change in height when moving horizontally on the surface, the work done is 0.
- **Conclusion**: Option 1 is true.

3. **Option 2: P on Earth's Surface and Q Above It**:
If point P is on the Earth's surface and point Q is above it, the work done will depend on the height difference between P and Q. The minimum work done will occur when the object is moved directly upward along the straight line from P to Q, as this minimizes the distance traveled against the gravitational force.
- **Conclusion**: This option is misleading because while the work done is indeed minimized along the straight line, it does not imply that the work done depends on the path taken. The work done is determined solely by the change in gravitational potential energy, which depends on the height difference.

4. **Option 3: Work Done Depends Only on Position**:
The work done in moving an object in a gravitational field is independent of the path taken and depends only on the initial and final positions. This is a fundamental principle in conservative forces like gravity.
- **Conclusion**: Option 3 is true.

5. **Option 4: No Work Done if Object is Returned to P**:
If the object is moved from P to Q and then back to P along any path, the initial and final positions are the same. Therefore, the net work done over the entire journey is 0, as the work done in moving to Q is equal and opposite to the work done in returning to P.
- **Conclusion**: Option 4 is true.

### Final Answer:
The true statements regarding the work done when moving an object in a gravitational field are:
- Option 1: True
- Option 3: True
- Option 4: True