Flywheel Angular Acceleration Calculation

Understanding the relationship between linear acceleration and angular acceleration is fundamental in rotational dynamics, particularly for a rigid body like a flywheel. This problem requires us to determine the angular acceleration of a flywheel given the linear acceleration of a point on its rim and its radius.

Acceleration Relationship Explained

For a point on a rigid body undergoing rotational motion, the linear acceleration (\(a\)) is directly proportional to the angular acceleration (\(\alpha\)) and the radial distance (\(r\)) from the axis of rotation to the point. The formula that connects these quantities is:

$$a = r \alpha$$

Where:

• \(a\) represents the linear acceleration of the point, measured in meters per second squared (m/s\(\text{^2}\)).
• \(r\) is the radius of the circular path followed by the point, measured in meters (m).
• \(\alpha\) is the angular acceleration of the rigid body, measured in radians per second squared (rad/s\(\text{^2}\)).

Given Values and Unit Conversion

First, let's identify the information provided in the problem statement:

• Linear acceleration (\(a\)) of a point on the flywheel = 3 m/s\(\text{^2}\)
• Radius of the flywheel (\(r\)) = 750 mm

To perform calculations using standard SI units, we need to convert the radius from millimeters (mm) to meters (m). Since 1 meter (m) equals 1000 millimeters (mm):

$$r = 750 \, \text{mm} \times \frac{1 \, \text{m}}{1000 \, \text{mm}}$$

$$r = 0.750 \, \text{m}$$

Calculating Angular Acceleration

Now, we can rearrange the relationship formula \(a = r \alpha\) to solve for the angular acceleration (\(\alpha\)):

$$\alpha = \frac{a}{r}$$

Substitute the given linear acceleration and the converted radius into this formula:

$$\alpha = \frac{3 \, \text{m/s}^2}{0.750 \, \text{m}}$$

Performing the division, we find the angular acceleration:

$$\alpha = 4 \, \text{rad/s}^2$$

Thus, the flywheel's angular acceleration is 4 rad/s\(\text{^2}\).