91Ó°ÊÓ

Understanding the moment of inertia is crucial when studying rotational dynamics. It is often referred to as the rotational equivalent of mass in linear motion. Moment of inertia measures an object's resistance to changes in its rotation. Just as heavier objects are harder to push in a straight line, objects with a larger moment of inertia are tougher to start or stop spinning.

For a uniform solid sphere, like the one in our exercise, the moment of inertia is calculated with the formula: \[ I = \frac{2}{5}mr^2 \]Here, \(m\) is the mass, and \(r\) is the radius of the sphere. By substituting the given values into this equation, we found the moment of inertia to be roughly \(3.245\, \text{kg m}^2\). This value tells us how much effort is needed to change the sphere's rotational state.

• Larger moment of inertia: More force needed to change rotation.
• Depends on mass distribution: Where the mass is located in relation to the axis of rotation.

Angular velocity plays a key role in the dynamics of rotating objects. It describes how fast an object rotates around an axis. It is measured in radians per second (rad/s). In our problem, angular velocity connects the sphere's kinetic energy with its moment of inertia.

We related the sphere's kinetic energy to its angular velocity using the formula:\[ KE = \frac{1}{2}I\omega^2 \]Solving for \(\omega\), the formula becomes:\[ \omega = \sqrt{\frac{2 \cdot KE}{I}} \]By plugging in the known kinetic energy and moment of inertia, we determined the angular velocity of the sphere to be about \(10.438\, \text{rad/s}\). Angular velocity helps us understand how quickly different parts of an object are moving.

• Higher angular velocity: Faster rotation.
• Directly affects tangential velocity at the object's edge.

Tangential velocity is the linear speed of a point at the edge of a rotating object. For a sphere or a circle, it describes how fast a point on the edge is moving along its circular path. This speed is directly connected to angular velocity, which we calculated earlier.

To find the tangential velocity \( v_{tan} \), we use the formula:\[ v_{tan} = r \omega \]where \(r\) is the radius of the sphere, and \(\omega\) is its angular velocity. Inserting our known values, we found the tangential velocity to be approximately \(3.97\, \text{m/s}\). This means a point on the rim of the sphere moves forward at this speed.

• Depends on both angular velocity and radius.
• Larger the radius, higher the tangential velocity for the same angular velocity.