A cable passes over a pulley. Because of friction, the tension in the cable is not eh same on opposite side of the pulley. The force in one side is 110 N, and the force on the other side is 100 N. Assume that the pulley is a uniform disk of mass 2.1kg and radius 0.21. Determine its angular acceleration.

To determine the angular acceleration of the pulley, we can use the principles of rotational dynamics. The net torque acting on the pulley is equal to the moment of inertia times the angular acceleration:

Net Torque = Moment of Inertia * Angular Acceleration

Now, let's break down the problem step by step:

1. Calculate the moment of inertia of the pulley:
The moment of inertia of a uniform disk rotating about its center is given by the formula:
I = (1/2) * m * r^2
where m is the mass of the disk and r is its radius. In this case, m = 2.1 kg and r = 0.21 m. Plugging these values into the formula, we get:

I = (1/2) * 2.1 kg * (0.21 m)^2 = 0.092 * kg * m^2

2. Calculate the net torque on the pulley:
The net torque is the difference between the torques exerted by the forces on the two sides of the pulley. Torque is given by the formula:
Torque = Force * Distance
where the distance is the radius of the pulley. In this case, the force on one side of the pulley is 110 N, and the force on the other side is 100 N. Plugging these values into the formula, we get:

Net Torque = (110 N * 0.21 m) - (100 N * 0.21 m) = 2.1 N * m

3. Solve for angular acceleration:
Using the equation from step 1 and the net torque from step 2, we can rearrange the equation to solve for the angular acceleration:

Net Torque = Moment of Inertia * Angular Acceleration
Angular Acceleration = Net Torque / Moment of Inertia

Plugging in the values we calculated, we get:

Angular Acceleration = 2.1 N * m / (0.092 * kg * m^2)

Simplifying, we find:

Angular Acceleration ≈ 22.83 rad/s^2

Therefore, the angular acceleration of the pulley is approximately 22.83 rad/s^2.