91Ó°ÊÓ

Torque is a measure of the rotational force applied at a distance from the axis of rotation. It plays a crucial role in understanding how forces can cause an object to rotate. In the context of the disk problem, torque arises from the frictional force acting at the edge of the disk.
To compute the torque (\( \tau \)), you use the formula:

• \( \tau = F_f \times r \)

In this case, \( F_f \) represents the frictional force, and \( r \) is the radius from the axis to where the force is applied.
To find the frictional force, multiply the coefficient of kinetic friction (\( \mu_k \)) by the normal force. The normal force here is identical to the weight of the disk:

• \( F_f = \mu_k \times \, \text{normal force} = 0.20 \times 5 = 1 \, \text{lb} \)
• \( \tau = 1 \, \text{lb} \times 0.25 \, \text{ft} = 0.25 \, \text{lb-ft} \)

This torque is what causes the disk to begin its rotational movement as it is pressed against the moving belt.

The Moment of Inertia (\( I \)) is a property that defines how difficult it is to change an object's rotational motion. Think of it as the rotational equivalent of mass in linear motion. For a symmetrical object like a disk, the Moment of Inertia takes into account its shape and mass distribution.
For a disk, the formula to calculate the Moment of Inertia is:

• \(I = \frac{1}{2} m r^2\)

In the exercise, first, convert the weight of the disk from pounds to mass using gravitational acceleration (32.2 ft/s²):

• \( m = \frac{5}{32.2} \, \text{slug} \)

Then, substitute the radius and the calculated mass into the formula:

• \( I = \frac{1}{2} \times \frac{5}{32.2} \times (0.25)^2 \)

This calculation gives the Moment of Inertia for the disk, a key value when determining how it reacts to the torque applied by the frictional force.

Kinematic equations allow us to describe the motion of objects without directly dealing with the forces causing this motion. Once we have the torque and moment of inertia, we can compute the angular parameters of the disk as it spins up to speed.
1. **Angular Acceleration (\( \alpha \))**: This is the rate at which the angular velocity changes. Using the relationship:

• \( \alpha = \frac{\tau}{I} \)

Substitute the known values of torque (\( \tau \)) and moment of inertia (\( I \)) calculated earlier to find \( \alpha \).
2. **Final Angular Velocity (\( \omega_f \))**: As the disk comes up to speed, it will eventually match the belt's linear velocity. Use:

• \( \omega_f = \frac{v}{r} = \frac{50}{0.25} \)

3. **Time to Reach Angular Velocity (\( t \))**: Finally, use the kinematic equation:

• \( \omega_f = \alpha \cdot t \)
• To solve for \( t \), rearrange to get \( t = \frac{\omega_f}{\alpha} \)

This equation provides the time required for the disk to reach its steady state of motion, driven by the angular acceleration \( \alpha \). With these tools, you can easily predict and understand the disk's behavior as it transitions to its constant angular velocity.