A disk with a mass of 14.0 kg, a diameter of 30.0 cm, and a thickness of 8.00 cm is mounted o

step 1 For this problem on the topic of rotation, we are given a disc which has a mass of 14 kilograms,

A disk with a mass of 14.0 kg, a diameter of 30.0 cm, and...

A disk with a mass of $14.0 \mathrm{~kg}$, a diameter of $30.0 \mathrm{~cm},$ and a thickness of $8.00 \mathrm{~cm}$ 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, $F=70.0 \mathrm{~N},$ is applied to the edge of the disk at an angle of $37.0^{\circ},$ as shown on the right in the figure. After $2.00 \mathrm{~s},$ the force is reduced to $F=24.0 \mathrm{~N},$ 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 2.00 s? c) What is the kinetic energy of the disk after $2.00 \mathrm{~s}$ ?

A disk with a mass of $14.0 \mathrm{~kg}$, a diameter of $30.0 \mathrm{~cm},$ and a thickness of $8.00 \mathrm{~cm}$ 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, $F=70.0 \mathrm{~N},$ is applied to the edge of the disk at an angle of $37.0^{\circ},$ as shown on the right in the figure. After $2.00 \mathrm{~s},$ the force is reduced to $F=24.0 \mathrm{~N},$ 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 2.00 s?
c) What is the kinetic energy of the disk after $2.00 \mathrm{~s}$ ?

Key Concepts

Constant Angular Velocity

Constant angular velocity refers to a state in which a rotating object maintains an unchanging rate of rotation over time. This occurs when the net external torque acting on the object is zero, meaning the driving torques are exactly balanced by resistive torques such as friction.

Frictional Torque

Frictional torque is the resistive torque that opposes the motion of a rotating object, typically arising from friction at the axle or contact surfaces. It reduces the net torque available to accelerate the object and is an important factor in systems where dissipative forces are present.

Rotational Kinetic Energy

Rotational kinetic energy is the energy possessed by a rotating object due to its motion. It is given by one half times the moment of inertia multiplied by the square of its angular velocity. This form of energy is essential when analyzing energy conservation and work done in rotational dynamics.

Moment of Inertia

The moment of inertia is a property of a body that quantifies its resistance to changes in its rotational motion about an axis. It depends on the mass distribution of the object with respect to the rotation axis, and it plays a central role in determining the angular acceleration produced by a given net torque.

Torque

Torque is a measure of the tendency of a force to rotate an object about an axis. It is calculated as the product of the force component perpendicular to the lever arm and the distance from the axis to the point of application of the force. This concept is crucial in analyzing how forces cause rotational acceleration in rigid bodies.

Angular Acceleration

Angular acceleration is the rate at which the angular velocity of a rotating object changes over time. It is directly proportional to the net torque acting on the system and inversely proportional to the moment of inertia. This relationship is the rotational analogue of Newton’s second law for linear motion.

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