A 30-cm-diameter, 1.2 kg solid turntable rotates on a 1.2-cm-diameter, 450 g shaft at a constant 33 rpm. When you hit the stop switch, a brake pad presses against the shaft and brings the turntable to a halt in 15 seconds. How much friction force does the brake pad apply to the shaft?

Verified step by step guidance

1. Step 1: Calculate the moment of inertia of the turntable. The turntable is a solid disk, so its moment of inertia is given by the formula: $I=\frac{1}{2}M{R}^2$, where $M$ is the mass of the turntable and $R$ is its radius. Convert the diameter to radius and use the given mass to compute the moment of inertia.
2. Step 2: Calculate the moment of inertia of the shaft. The shaft is also a solid cylinder, so its moment of inertia is given by the formula: $I=\frac{1}{2}M{R}^2$. Use the given diameter to find the radius and the mass of the shaft to compute its moment of inertia.
3. Step 3: Determine the total moment of inertia of the system. Add the moment of inertia of the turntable and the shaft together: $I=I$_turntable+I_shaft.
4. Step 4: Calculate the angular deceleration. The turntable slows down from an initial angular velocity of 33 rpm to 0 rpm in 15 seconds. Convert the angular velocity from rpm to rad/s using the conversion factor $\left(\frac{2}{\pi }\right)$ and divide by 60. Then use the formula for angular deceleration: $\alpha =\frac{\Delta }{\omega }t$, where $\Delta \omega$ is the change in angular velocity and $t$ is the time.
5. Step 5: Calculate the friction force. The torque due to friction is related to the angular deceleration by the formula: $\tau =I\alpha$. The torque is also related to the friction force by the formula: $\tau =F$_frictionR, where $R$ is the radius of the shaft. Combine these equations to solve for the friction force: $F$_friction=IαR.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object, calculated as the product of the force and the distance from the pivot point (lever arm). In this scenario, the brake pad applies a torque to the shaft, which is responsible for decelerating the turntable. Understanding torque is essential for analyzing how forces affect rotational motion.

Moment of Inertia

Moment of inertia quantifies an object's resistance to changes in its rotational motion, depending on its mass distribution relative to the axis of rotation. For the turntable and shaft, calculating the moment of inertia is crucial to determine how much torque is needed to stop the turntable. It plays a key role in the dynamics of rotating systems.

Friction Force

Friction force is the resistance encountered when one surface moves over another, and it is essential for stopping motion. In this case, the brake pad generates a friction force against the shaft to bring the turntable to a halt. The magnitude of this force can be calculated using the torque and the radius of the shaft, illustrating the relationship between linear and rotational dynamics.