The center of mass is the point where we can consider all of the substance's mass to be concentrated. Finding the center of mass greatly facilitates mechanics problems, because we can perceive the force to be acting on only this one point.

Since the lead side of the cylinder is heavier than the aluminum side, we can conclude that the center of mass is located inside the lead side of the cylinder. Since the lead side has a mass of 60g, each inch of the lead side will have a mass of 15g. By going one inch into the lead side, there are 45g on each side of the point. As a result, we say that this is the location for the center of mass for the cylinder, as either side would weigh the same individually.

A body achieves rotational equilibrium if it is rotating at a constant rate (or no rate, if it is at rest). Similarly, a body achieves translational equilibrium if it is moving at a constant rate (or no rate at all).

In order to impact translational equilibrium, an external force must be applied (Newton's first law). This force results in the acceleration of the mass (Newton's second law).

In order to impact rotational equilibrium, an external torque must be applied. This torque results in rotational acceleration by introducing a centripetal force.

An object can be in rotational equilibrium without translational equilibrium, and can be in translational equilibrium without rotational equilibrium. These two factors are independent in most cases.

Due to conservation of angular momentum, the angular momentum L must remain constant: .

Because of the child’s decreased distance from the axis of rotation, the moment of inertia of the system also decreases The child can be approximated as a point mass with moment of inertia , and the carousel’s moment of inertia remains unchanged. Moving the child toward the center of the carousel will decrease its moment of inertia.

To examine what happens to angular velocity, we need the definition of angular momentum, . Replacing this in the conservation of angular momentum equation, we see that . Since , then  so that the product remains the same.

Angular momentum remains the same, moment of inertia decreases, and angular velocity increases. 

We can apply the right-hand rule for angular velocity: if the fingers on the right hand curl in the direction of rotation, the thumb points in the direction of the angular velocity vector. In this case the fingers curl clockwise, so thumb points downwards.

The angular acceleration would decrease. Newton’s second law states that F = ma. The net force, however, is reduced in the internal friction scenario because the friction acts to oppose the motion created by the rope around the pulley. As a side note, the pulley would also heat up because the energy change due to friction is released as heat.

The equation relating velocity and radius is  .

Because angular velocity can also be found using the equation , we can set the equations equal and solve for the velocity.

Since we are given the radius as  and the frequency as , we can calculate the velocity. (Frequency can be found by the number of times an object travels around a full circle in one second).

Alternatively, we can use the radius to determine the circumference of the collider.

The particle travels twice around the collider every second.

An object with low moment of inertia will acquire less rotational kinetic energy and more translational kinetic energy (moves faster down the ramp) than an object with higher moment of inertia.

Moment of inertia depends on how close to the object's center of mass its mass is concentrated. An object with more of its mass close to the center will have lower moment of inertia, and subsequently more translational kinetic energy. The solid sphere has lowest moment of inertia, then the disk, and then the hollow sphere, thus the solid sphere reaches the end first, then the disk, and finally the hollow sphere.

Since the ramp is frictionless all the energy in the system is conserved. All the potential energy is converted into kinetic energy.

The potential energy is given by: 

The kinetic energy formula is:

Since all of the potential energy is converted to kinetic energy we can solve for velocity.