Rotating Systems: Energy & Momentum

What is the total kinetic energy of a system that has both translational and rotational motion?

An object with mass $m = 2 \text{ kg}$ is moving at a velocity $v = 3 \text{ m/s}$ and rotating with an angular velocity $\omega = 4 \text{ rad/s}$. If its moment of inertia is $I = 0.5 \text{ kg m}^2$, what is its total kinetic energy?

A sphere rolls down an inclined plane. As it descends, its potential energy is converted into both translational and rotational kinetic energy. If the sphere's initial potential energy is $100 \text{ J}$, and at some point its translational kinetic energy is $60 \text{ J}$, what is its rotational kinetic energy at that...

For an object rolling without slipping, what is the relationship between its linear velocity ($v$) and angular velocity ($\omega$) if the radius of the object is $r$?

A wheel with a radius of $0.5 \text{ m}$ is rolling without slipping. If its angular velocity is $6 \text{ rad/s}$, what is its linear velocity?

A solid cylinder rolls without slipping down an inclined plane. Which of the following affects the acceleration of the cylinder?

What is the primary difference between rolling with slipping and rolling without slipping in terms of energy?

A wheel is rolling with slipping on a surface. If the kinetic friction force is $5 \text{ N}$ and the distance over which it slips is $2 \text{ m}$, how much energy is lost due to friction?

A ball is rolling with slipping. How does the work done by kinetic friction affect the mechanical energy of the system?

A block is sliding down a ramp while a cylinder rolls without slipping down an identical ramp. Both have the same mass and start from the same height. Which reaches the bottom first?