A solid cylinder is rolling without slipping down an inclined plane. Then its angular momentum is :

Options

A. Conserved about COM of the cylinder
B. Conserved about point of contact
C. Conserved about all the points
D. Not conserved about any point

Answer

To determine the angular momentum of a solid cylinder rolling without slipping down an inclined plane, we can analyze the situation step by step.

### Step 1: Understand the System
A solid cylinder is rolling down an inclined plane. The forces acting on the cylinder include gravitational force (mg), normal force, and frictional force. The cylinder rolls without slipping, which means there is a relationship between its linear and angular motion.

**Hint:** Identify the forces acting on the cylinder and their directions.

### Step 2: Identify the Torque
The torque (τ) acting on the cylinder can be calculated using the formula:
\[ \tau = r \times F \]
where \( r \) is the radius of the cylinder and \( F \) is the force causing the torque. In this case, the force is the component of gravitational force acting down the incline, which is \( mg \sin \theta \).

**Hint:** Remember that torque is the product of the radius and the force applied perpendicular to the radius.

### Step 3: Calculate the Torque
The torque due to the gravitational force acting down the incline is given by:
\[ \tau = r \cdot (mg \sin \theta) \]
This torque causes the angular momentum of the cylinder to change.

**Hint:** Use the relationship between torque and angular momentum, \( \tau = \frac{dL}{dt} \).

### Step 4: Relate Torque to Angular Momentum
From the relationship \( \tau = \frac{dL}{dt} \), we can see that if there is a non-zero torque, the angular momentum \( L \) is changing with time. This means that the angular momentum is not conserved.

**Hint:** If torque is present, angular momentum cannot be conserved.

### Step 5: Conclusion
Since the torque acting on the cylinder is not zero (due to the gravitational force component along the incline), the angular momentum of the cylinder is not conserved about any point.

**Final Answer:** The correct option is D: not conserved about any point.