A stationary bicycle is raised off the ground, and its front...

A stationary bicycle is raised off the ground, and its front wheel $(m=1.3 \mathrm{kg})$ is rotating at an angular velocity of $13.1 \mathrm{rad} / \mathrm{s}$ (see the drawing). The front brake is then applied for $3.0 \mathrm{s},$ and the wheel slows down to $3.7 \mathrm{rad} / \mathrm{s} .$ Assume that all the mass of the wheel is concentrated in the rim, the radius of which is $0.33 \mathrm{m} .$ The coefficient of kinetic friction between each brake pad and the rim is $\mu_{\mathrm{k}}=0.85 .$ What is the magnitude of the normal force that each brake pad applies to the rim?

A stationary bicycle is raised off the ground, and its front wheel $(m=1.3 \mathrm{kg})$ is rotating at an angular velocity of $13.1 \mathrm{rad} / \mathrm{s}$ (see the drawing).
The front brake is then applied for $3.0 \mathrm{s},$ and the wheel slows down to $3.7 \mathrm{rad} / \mathrm{s} .$ Assume that all the mass of the wheel is concentrated in the rim, the radius of which is $0.33 \mathrm{m} .$ The coefficient of kinetic friction between each brake pad and the rim is $\mu_{\mathrm{k}}=0.85 .$ What is the magnitude of the normal force that each brake pad applies to the rim?

Key Concepts

Normal Force Distribution in Brake Systems

In systems employing multiple brake pads, the total frictional force is the sum of the forces provided by each pad. Therefore, when calculating the normal force applied by each brake pad, it is important to consider how the braking torque is distributed between them. This concept helps in understanding how individual forces contribute to the overall braking effect on a rotating wheel.

Rotational Dynamics

Rotational dynamics involves the relationship between torque, moment of inertia, and angular acceleration, known as Newton’s second law for rotation (? = I*?). This principle is critical in problems where a net torque (from friction, in this case) is responsible for the change in a spinning object's angular velocity over time.

Frictional Torque

Frictional torque is generated by frictional forces acting at a distance (the lever arm) from the axis of rotation. In braking systems, the friction between the brake pads and the rim creates a torque that opposes the wheel’s rotation. The frictional force itself is determined by the coefficient of kinetic friction multiplied by the normal force, and when combined with the radius at which the force acts, it yields the braking torque.

Moment of Inertia

The moment of inertia is a measure of an object's resistance to changes in its rotational motion and depends on how the object’s mass is distributed relative to the axis of rotation. For a hoop or rim, the moment of inertia is given by I = m*r². In problems involving rotational deceleration, knowing the moment of inertia is essential since it, coupled with angular acceleration, allows the computation of the net torque acting on the system.

Angular Deceleration

Angular deceleration describes the rate at which an object's angular velocity decreases over time. It is calculated by finding the difference between the final and initial angular velocities and dividing by the time interval. This concept is key for determining how quickly a rotating body, such as a bicycle wheel, slows down when subjected to a braking force.

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