A 2.1-kg 15-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 14 N.
(a) Find the magnitude of the torque exerted by the rope.
?N · m
(b) Find the angular acceleration of the cylinder.
?rad/s2
(c) Find the angular velocity of the cylinder at t = 1.00 s.
?rad/s
To answer these questions, we need to use the concepts of torque, angular acceleration, and angular velocity. Let's break down each question and explain how to find the answers.
(a) To find the magnitude of the torque exerted by the rope, we can use the formula:
Torque = Force * Lever Arm
The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force. In this case, the lever arm is equal to the radius of the cylinder. Substitute the given values into the formula:
Torque = 14 N * 0.15 m
Torque = 2.1 N·m
Therefore, the magnitude of the torque exerted by the rope is 2.1 N·m.
(b) To find the angular acceleration of the cylinder, we can use the formula:
Torque = Moment of Inertia * Angular Acceleration
The moment of inertia depends on the shape and mass distribution of the object. For a cylindrical object rotating about its axis, the formula is:
Moment of Inertia = (1/2) * Mass * Radius^2
Substitute the given values into the formula:
Moment of Inertia = (1/2) * 2.1 kg * (0.15 m)^2
Moment of Inertia = 0.023625 kg·m^2
Now, rearrange the formula to solve for angular acceleration:
Angular Acceleration = Torque / Moment of Inertia
Angular Acceleration = (2.1 N·m) / (0.023625 kg·m^2)
Angular Acceleration = 88.89 rad/s^2
Therefore, the angular acceleration of the cylinder is 88.89 rad/s^2.
(c) To find the angular velocity of the cylinder at t = 1.00 s, we can use the formula:
Final Angular Velocity = Initial Angular Velocity + (Angular Acceleration * Time)
Given that the initial angular velocity is 0 rad/s (since the cylinder is initially at rest), and the angular acceleration is 88.89 rad/s^2, substitute these values into the formula:
Final Angular Velocity = 0 rad/s + (88.89 rad/s^2 * 1.00 s)
Final Angular Velocity = 88.89 rad/s
Therefore, the angular velocity of the cylinder at t = 1.00 s is 88.89 rad/s.