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.

There are two types of units: base units and derived units. Base units are a set of units from which all other units are derived, whereas derived units are the units derived from base units.

There are six main base units: meters, kilogram, second, Ampere, Kelvin, moles, and candela (unit for luminous intensity). The rest of the units are classified as derived units and will contain a combination of these base units. The SI units for the four measurements listed in question are as follows:

Mass: 

Volume:  (Liters)

Weight:  (Newtons)

Density: 

This means that volume, weight, and density are derived units, whereas mass is a base unit. 

Derived units are derived from the six base units. A derived unit usually contains multiple base units that are combined by using multiplication and/or division; addition and subtraction of base units is not performed to obtain derived units.

The SI unit for force is Newtons. Newtons written in base units is ; therefore, units for force is always a derived unit. A unit can be either a derived unit or a base unit, but it can never be both. Recall that there are only six base units and numerous derived units; therefore, there are more derived units than base units. 

The SI base unit for time is seconds. Hours are derived by multiplying seconds by sixty. Hours are not a base unit.

Recall that the mass of a substance is usually measured in kilograms (), which is a type of base unit. Moles () is a unit used to measure the amount of substance. Moles are also a type of base unit; therefore, both the unit for mass () and the unit for amount of substance () are measures of base units. 

Scalar quantities are those that can be described with magnitude only, as opposed to vectors, which include both magnitude and direction components. Distance, speed, and time are all scalars. Displacement is not a scalar, as it involves both the distance and the direction moved from a starting point. Velocity also includes a direction component, and is therefore a vector quantity.