A disk with a mass of 17 kg, a diameter of 25 cm, and a thickness of 8 cm is mounted on a rough horizontal axle as shown on the left in the figure. (There is a friction force between the axle and the disk.) The disk is initially at rest. A constant force, F = 70 N, is applied to the edge of the disk at an angle of 37°, as shown on the right in the figure. After 2.0 s, the force is reduced to F = 17 N, and the disk spins with a constant angular velocity.
(a) What is the magnitude of the torque due to friction between the disk and the axle?
(b) What is the angular velocity of the disk after 2.0 s?
(c) What is the kinetic energy of the disk after 2.0 s?
To solve this problem, we will use the principles of rotational dynamics. Let's break down each part of the question and find the required answers.
(a) To find the magnitude of the torque due to friction between the disk and the axle, we need to use the formula for torque:
Torque = Force x Distance x sin(θ)
In this case, the distance is the radius of the disk, which is half the diameter: r = 25 cm / 2 = 12.5 cm = 0.125 m.
The angle between the force and the distance is 90° because the force is applied tangentially at the edge of the disk. Hence, sin(θ) = sin(90°) = 1.
Using the given constant force F = 70 N, we can calculate the torque:
Torque = 70 N x 0.125 m x 1 = 8.75 Nm
So, the magnitude of the torque due to friction between the disk and the axle is 8.75 Nm.
(b) To find the angular velocity of the disk after 2.0 s, we need to use the equation for angular acceleration:
Angular acceleration = Torque / Moment of inertia
The moment of inertia of a thin disk rotating about its axis is given by:
Moment of inertia (I) = (1/2) * m * r^2
where m is the mass of the disk (17 kg) and r is the radius of the disk (0.125 m).
Plugging in the values, we get:
I = (1/2) * 17 kg * (0.125 m)^2 = 0.133 kg * m^2
We are given that the force is reduced to F = 17 N. Using this force, we can calculate the torque:
Torque = 17 N * 0.125 m * 1 = 2.125 Nm
Now, we can calculate the angular acceleration:
Angular acceleration = Torque / Moment of inertia = 2.125 Nm / 0.133 kg * m^2
After 2.0 s, the angular velocity will be:
Angular velocity = Initial angular velocity + (Angular acceleration * time)
Since the disk is initially at rest, the initial angular velocity is zero:
Angular velocity = (Angular acceleration * time)= (2.125 Nm / 0.133 kg * m^2) * 2.0 s
Simplifying this expression will give us the angular velocity of the disk after 2.0 s.
(c) To find the kinetic energy of the disk after 2.0 s, we use the formula:
Kinetic energy = (1/2) * I * (angular velocity)^2
Here, we already know the moment of inertia (0.133 kg * m^2) and the angular velocity (calculated in part b).
Plugging these values into the formula, we can find the kinetic energy of the disk after 2.0 s.