A 2.9-kg 12-cm-radius cylinder, initially at rest, is free to rotate about the axis of the cylinder. A rope of negligible mass is wrapped around it and pulled with a force of 20 N.

(a) Find the magnitude of the torque exerted by the rope.

(b) Find the angular acceleration of the cylinder.

(c) Find the angular velocity of the cylinder at t = 0.90 s.

To find the magnitude of the torque exerted by the rope on the cylinder (part a), we can use the formula:

Torque = Force * Lever Arm

where the lever arm is the perpendicular distance from the axis of rotation to the point where the force is applied.

In this case, the force is 20 N, and the lever arm would be the radius of the cylinder, which is 12 cm or 0.12 m.

Thus, the torque exerted by the rope would be:

Torque = Force * Lever Arm
= 20 N * 0.12 m
= 2.4 N*m

So, the magnitude of the torque exerted by the rope is 2.4 N*m.

To find the angular acceleration of the cylinder (part b), we can use Newton's second law for rotation, which states:

Torque = Moment of Inertia * Angular Acceleration

The moment of inertia for a solid cylinder rotating about its axis is given by:

Moment of Inertia = (1/2) * Mass * Radius^2

In this case, the mass of the cylinder is 2.9 kg, and the radius is 0.12 m. Plugging these values into the equation, we get:

Moment of Inertia = (1/2) * 2.9 kg * (0.12 m)^2
= 0.04176 kg*m^2

Now we can rearrange the equation to solve for angular acceleration:

Angular Acceleration = Torque / Moment of Inertia
= 2.4 N*m / 0.04176 kg*m^2
= 57.59 rad/s^2

So, the angular acceleration of the cylinder is 57.59 rad/s^2.

To find the angular velocity of the cylinder at t = 0.90 s (part c), we can use the formula for angular acceleration:

Angular Acceleration = (Final Angular Velocity - Initial Angular Velocity) / Time

In this case, the initial angular velocity is 0 since the cylinder is initially at rest. We need to solve for the final angular velocity.

Rearranging the equation, we get:

Final Angular Velocity = Initial Angular Velocity + (Angular Acceleration * Time)

Since the initial angular velocity is 0, the equation simplifies to:

Final Angular Velocity = Angular Acceleration * Time

Plugging in the values, we get:

Final Angular Velocity = 57.59 rad/s^2 * 0.90 s
= 51.83 rad/s

So, the angular velocity of the cylinder at t = 0.90 s is 51.83 rad/s.

I hope this helps! Let me know if you have any further questions.