HiSET Science: Conservation of Energy and Momentum

Applying Conservation of Energy and Momentum
A metal hoop of mass m and radius r is released from rest at the top of a hill of height h. Assuming that it rolls without sliding and does not lose energy to friction or drag, what will be the hoop's angular and linear velocities upon reaching the bottom of the hill?
The hoop's initial energy is all potential energy, PE = mgh. As the hoop rolls down, all of its energy is converted to translational and rotational kinetic energy. Thus, mgh = mv2/2 + Iω2/2.
Since the moment for a hoop is I = mr2, and ω = v/r, the equation may be rewritten as mgh = mv2/2 + mr2(v2/r2)/2, which further simplifies to mgh = mv2. Thus, the resulting velocity of the hoop is vf = sqrt(gh), with an angular velocity of ωf= vf/r. Note that if you were to forget about the energy converted to rotational motion, you would calculate a final velocity of vf = sqrt(2gh), which is the impact velocity of an object dropped from height h.
Consider a planet orbiting the sun through an elliptical orbit with small radius rS and large radius rL. Find the angular velocity of the planet when it is at distance rS from the sun if its velocity at rL is ωL.
Since the size of a planet is almost insignificant compared to the interplanetary distances, the planet may be treated as a single particle of mass m, giving it a moment about the sun of I = mr2. Since the gravitational force is incapable of exerting a net torque on an object, we can assume that the planet's angular momentum about the sun is a constant. Thus, mrL2ωL= mrS2ωS. Solving this equation for ωS yields ωS = ωL(rL2/rS2).

Mass-Energy Relationship
Because mass consists of atoms, which are themselves formed of subatomic particles, there is an energy inherent in the composition of all mass. If all the atoms in a given mass were formed from their most basic particles, it would require a significant input of energy. This rest energy is the energy that Einstein refers to in his famous mass-energy relation E = mc2, where c is the speed of light in a vacuum. In theory, if all the subatomic particles in a given mass were to spontaneously split apart, it would give off energy E = mc2. For example, if this were to happen to a single gram of mass, the resulting outburst of energy would be E = 9 x 1013 J, enough energy to heat more than 200,000 cubic meters of water from the freezing point to the boiling point.
In some nuclear reactions, small amounts of mass are converted to energy. The amount of energy released can be calculated through the same relation, E = mc2. Most such reactions involve mass losses on the order of 10-30 kg.