Problem:
A solid homogeneous cylinder of height h = 1 m and base radius r = 1 m is kept vertically on a conveyor belt moving horizontally with an increasing velocity v = t^2. If the cylinder is not allowed to slip, find the time when the cylinder is about to topple. The correct answer is '10'. Explain this answer in detail.

Solution:

To understand when the cylinder is about to topple, we need to consider the forces acting on it and analyze the conditions for equilibrium.

1. Forces Acting on the Cylinder:
When the cylinder is about to topple, the net torque acting on it must be zero. The forces acting on the cylinder are:
- Weight (mg) acting downward
- Normal force (N) exerted by the conveyor belt
- Frictional force (f) between the cylinder and the conveyor belt

2. Conditions for Equilibrium:
For the cylinder to be in equilibrium, the following conditions must be satisfied:
- Net force in the horizontal direction = 0
- Net torque about the point of contact with the conveyor belt = 0

3. Analysis:
Let's analyze the forces acting on the cylinder at a general time 't'.

- Weight (mg): The weight of the cylinder acts downward and can be calculated as mg = m * g, where m is the mass of the cylinder and g is the acceleration due to gravity.

- Normal force (N): The normal force exerted by the conveyor belt opposes the weight of the cylinder and acts upward. At equilibrium, N = mg.

- Frictional force (f): The frictional force between the cylinder and the conveyor belt acts in the opposite direction of the cylinder's motion. It can be calculated as f = μN, where μ is the coefficient of friction.

- Net force in the horizontal direction: The net force in the horizontal direction is given by F_net = f - m * a, where a is the acceleration of the cylinder.

- Net torque about the point of contact: The net torque about the point of contact with the conveyor belt is given by τ_net = r * f - r * m * a, where r is the radius of the cylinder.

4. Equilibrium Conditions:
Using the conditions for equilibrium, we can set up the following equations:

- Net force in the horizontal direction = 0:
f - m * a = 0

- Net torque about the point of contact = 0:
r * f - r * m * a = 0

5. Solving the Equations:
Substituting the expression for f from the first equation into the second equation, we get:
r * μN - r * m * a = 0

Substituting N = mg and rearranging the equation, we have:
r * μ * mg - r * m * a = 0

Simplifying the equation, we get:
μ * g - a = 0

Rearranging the equation, we have:
a = μ * g

6. Finding the Time When the Cylinder is about to Topple:
We are given that the velocity of the conveyor belt is v = t^2. Acceleration is the derivative of velocity with respect to time, so we differentiate the equation for velocity:
a