91Ó°ÊÓ

When dealing with circular motion, it is essential to understand how to convert linear speed into angular speed. Linear speed is the speed along a straight path, whereas angular speed is the rate at which an object moves along a circular path. The relationship between linear speed and angular speed is given by the formula:

• \( v = r\omega \)

Here, \( v \) represents linear speed, \( r \) is the radius of the circular path, and \( \omega \) denotes angular speed.

Given the linear speed and the radius, you can easily find the angular speed by rearranging the formula to:

• \( \omega = \frac{v}{r} \)

This conversion is pivotal in problems involving circular motion, such as analyzing the dynamics of a grinding wheel. By understanding this relation, you can seamlessly transition between linear and angular quantities, which simplifies broader calculations such as determining angular acceleration.

To calculate angular speed, we must first determine the radius of the circular path. The grinding wheel in our problem has a diameter of \( 0.75 \, \text{m} \), so its radius is:

• \( r = \frac{0.75}{2} = 0.375 \, \text{m} \)

Using the radius and the given linear speeds, we can find the initial and final angular speeds. For the initial linear speed of \( 12\, \text{m/s} \), the initial angular speed \( \omega_i \) is computed as follows:

• \( \omega_i = \frac{12}{0.375} = 32 \, \text{rad/s} \)

Similarly, the final angular speed \( \omega_f \) for the linear speed of \( 25\, \text{m/s} \) is:

• \( \omega_f = \frac{25}{0.375} = 66.67 \, \text{rad/s} \)

This step transitions the analysis from linear to angular measurements, critical for subsequent calculations like finding angular acceleration.

Understanding the dynamics of a grinding wheel involves analyzing how its angular speed changes over time. The angular acceleration, \( \alpha \), is a key component here. It describes the rate of change of angular speed and is calculated by dividing the change in angular speed by the time interval over which the change occurs:

• \( \alpha = \frac{\Delta \omega}{\Delta t} \)

In our example, the change in angular speed \( \Delta \omega \) is:

• \( \Delta \omega = \omega_f - \omega_i = 66.67 - 32 = 34.67 \, \text{rad/s} \)

Given the time duration \( \Delta t \) of \( 6.2 \, \text{s} \), the average angular acceleration is computed as:

• \( \alpha = \frac{34.67}{6.2} \approx 5.59 \, \text{rad/s}^2 \)

The average angular acceleration provides insight into how the rotational speed of the wheel is increasing and is a vital metric for understanding the wheel's response to different forces and torques.