Torque and Rotational Motion

According to Newton's Second Law for Rotation, what is the relationship between net torque ($sum au$), rotational inertia ($I$), and angular acceleration ($\alpha$)?

A wheel of radius $R$ rolls without slipping along a horizontal surface. If the angular speed of the wheel is $\omega$, what is the linear speed of its center of mass?

A wheel with a rotational inertia of $2.0 \ kg cdot m^2$ experiences a net torque of $10.0 \ N cdot m$. What is its angular acceleration?

Two torques act on a rotating disc. Torque $\tau_1 = 5 \ N cdot m$ acts clockwise, and torque $\tau_2 = 12 \ N cdot m$ acts counterclockwise. If the disc has a rotational inertia of $4 \ kg cdot m^2$, what is the magnitude of the angular acceleration?

A complex machine part consists of a thin rod of mass $m$ and length $L$ attached to a solid sphere of mass $2m$ and radius $R = L/4$ at the end of the rod. The system rotates about an axis at the other end of the rod. If a net torque $\tau$ is applied to this system, what is the angular acceleration, given that $I_{rod} = \frac{1}{3}mL^2$ and $I_{sphere} = \frac{2}{5}mR^2 + mD^2$?

Two torques are applied to an object. Torque $\tau_1$ has a magnitude of $5 \ N cdot m$ and points along the +z axis, while torque $\tau_2$ has a magnitude of $3 \ N cdot m$ and points along the -z axis. If the object's rotational inertia is $2 \ kg cdot m^2$, what is the direction of the angular acceleration?

A figure skater is spinning with her arms extended. She then pulls her arms in closer to her body. What happens to her angular speed?

A point on a rotating object is at a distance of $0.5 \ m$ from the axis of rotation. If the object has an angular speed of $4 \ rad/s$, what is the tangential speed of the point?

A clockwise torque is applied to a stationary wheel. In which direction will the angular acceleration of the wheel point?

A solid cylinder and a hollow cylinder have the same mass and radius. Which one has a greater rotational inertia?