When there is no external torque acting on a body moving in an elliptical path, which of the following quantities remain constant ?

Options

A. Kinetic energy
B. Potential energy
C. Linear momentum
D. Angular momentum

Answer

To solve the problem of which quantities remain constant when a body is moving in an elliptical path with no external torque acting on it, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Torque and Angular Momentum**:
- When there is no external torque acting on a body, it implies that the net torque \( \tau \) is zero.
- According to the rotational dynamics, if the net torque is zero, the rate of change of angular momentum \( L \) is also zero:
\[ \frac{dL}{dt} = \tau = 0 \]
- This means that the angular momentum \( L \) remains constant.

2. **Analyzing Other Quantities**:
- Next, we need to analyze whether other quantities like kinetic energy and potential energy remain constant.
- The kinetic energy \( KE \) of the body is given by:
\[ KE = \frac{1}{2} mv^2 \]
- As the body moves in an elliptical path, its speed \( v \) changes due to the varying distance from the focus of the ellipse, thus the kinetic energy is not constant.

3. **Potential Energy Consideration**:
- The potential energy \( U \) in a gravitational field can be expressed as:
\[ U = -\frac{G m_1 m_2}{r} \]
- As the distance \( r \) between the two masses changes during the elliptical motion, the potential energy also changes. Therefore, potential energy is not constant.

4. **Conclusion**:
- From the analysis above, we conclude that the only quantity that remains constant when a body moves in an elliptical path with no external torque is the angular momentum \( L \).

### Final Answer:
The quantity that remains constant is **angular momentum**.

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