To find the linear velocity of the rim of the large pulley, we use the following relationship:

$\text{Linear Velocity} = \text{Angular Velocity} \times \text{Radius}$

Here, the angular velocity is given in revolutions per minute (rpm), and the radius is given in centimeters. To express the velocity in inches per minute, we first convert the radius from centimeters to inches.

Step 1: Convert the radius to inches

Given:

• Radius of the large pulley, $r = 24$ centimeters.

To convert centimeters to inches, use the conversion factor:

$1 \text{ cm} = 0.3937 \text{ inches}$

Thus:

$r = 24 \times 0.3937 = 9.4488 \text{ inches}$

Step 2: Calculate the linear velocity

The angular velocity is given as $75$ revolutions per minute.

The linear velocity $v$ is calculated as:

$v = \omega \times r$

where:

• $\omega = 75$ rpm (angular velocity),
• $r = 9.4488$ inches.

Now, since 1 revolution covers a distance equal to the circumference of the pulley, which is $2\pi r$ inches, we have:

$v = 75 \times 2\pi \times 9.4488 \text{ inches per minute}$

Step 3: Compute the result

Let's compute this:

$v = 75 \times 2 \times 3.1416 \times 9.4488$ $v \approx 75 \times 59.3613$ $v \approx 4452.0975 \text{ inches per minute}$

Final Answer

The linear velocity of the rim of the large pulley is approximately 4452.10 inches per minute.

Would you like more details on any of the steps or have any questions?

---

Here are some related questions you might be interested in:

1. What is the angular velocity of the smaller pulley?
2. How does the radius of the pulley affect the linear velocity?
3. What would happen to the linear velocity if the motor speed increased?
4. How can you calculate the linear velocity if the radii and rpm of both pulleys are unknown?
5. How would the direction of rotation change if the belt crossed between the pulleys?

Tip: Remember that linear velocity increases directly with both the angular velocity and the radius of the pulley.