91Ó°ÊÓ

The moment of inertia is a fundamental concept in understanding rotational motion. It reflects how the mass of an object is distributed with respect to its axis of rotation. The formula for the moment of inertia depends on the shape of the object and the axis about which it rotates. In our example, we have a solid sphere rotating about an axis through its center. The formula to calculate its moment of inertia is given by \[ I = \frac{2}{5} m r^2 \]where:

• \( m = 0.225 \, \text{kg} \) - the mass of the sphere
• \( r = 0.015 \, \text{m} \) - the radius of the sphere

Substitute the known values into the formula to find:\[ I = \frac{2}{5} \times 0.225 \times (0.015)^2 = 2.025 \times 10^{-5} \, \text{kg} \cdot \text{m}^2 \]This value tells us how hard or easy it is for the sphere to change its rotational speed. The larger the moment of inertia, the more torque is needed to achieve the same angular acceleration.

Torque is essentially the rotational equivalent of linear force. It's what causes an object to start spinning or change its rate of spin. To find the torque, we multiply the force applied by the distance from the axis of rotation, which is referred to as the lever arm. In our scenario:

• The friction force \( F = 0.0200 \, \text{N} \)
• The radius or lever arm \( r = 0.015 \, \text{m} \)

Now apply the torque formula: \[ \tau = F \times r \] Substituting the given values yields:\[ \tau = 0.0200 \times 0.015 = 3.00 \times 10^{-4} \, \text{N} \cdot \text{m} \]Torque, therefore, is dependent on both the force applied and the distance from the axis at which it's applied. A higher torque results in higher angular acceleration, assuming the moment of inertia stays constant.

Angular velocity is a measure of how fast something is rotating. It's similar to linear velocity, but instead of units of distance over time (like meters per second), angular velocity is expressed in radians per second (rad/s). When we discuss changing angular velocity, we often refer to angular acceleration, which is the rate of change of angular velocity.In our example, we need to find the time it takes for the sphere's rotational speed to decrease by a certain amount (\( 22.5 \, \text{rad/s} \)). The relationship between angular acceleration \( \alpha \), change in angular velocity \( \Delta \omega \), and time \( t \) can be expressed as:\[ \Delta \omega = \alpha t \]Here:

• \( \Delta \omega = 22.5 \, \text{rad/s} \) - the desired change in angular velocity
• \( \alpha = 14.82 \, \text{rad/s}^2 \) - the angular acceleration we calculated earlier

We use the formula to solve for time:\[ t = \frac{\Delta \omega}{\alpha} = \frac{22.5}{14.82} \approx 1.52 \, \text{seconds} \]This calculation shows how long it will take for the frictional force to bring the sphere's rotation down by \( 22.5 \, \text{rad/s} \), demonstrating the practical effects of angular velocity and acceleration in everyday applications.