Gravitational Potential Energy When we are close to the surface of the Earth we use the constant value of g. If we are at some altitude above the surface of the Earth we must use the radially dependent form of the gravitational acceleration. Assumes constant g! 0 0 We will define U i = 0 when r i = For more than two masses: Only valid for r > R E !

Energy Considerations for Planetary Motion This is the total energy of an object orbiting a larger stationary mass. This is a reasonable assumption if m << M. For circular orbits, the speed of the small mass is constant. For elliptical orbits the speed of the small mass is not constant! For a circular orbit we can determine v in the following way: 1 2 What happens as the planet moves from 1 to 2? What happens as the planet moves from 2 to 1? The distance from the sun decreases and hence the potential energy decreases, therefore the kinetic energy and hence the speed must increase. The distance from the sun increases and hence the potential energy increases, therefore the kinetic energy and hence the speed must decrease. We can now rewrite the energy:

Escape Speed The term escape speed or escape velocity if often used when discussing the launching of a rocket into space. What is escape speed? Escape speed is the minimum initial speed required for an object to approach an infinite separation from the Earth. There is no propulsion system on this object. 0

Black Holes What is a black hole? Gravitational well with extremely high gravitational forces. Singularity Event Horizon (Point of no return) RsRs Creation of a black hole: 1.Massive star explodes – Supernova 2.Mass that does not travel far enough gets pulled back and the mass collapses under gravitational forces to form one of the following: a)White Dwarf – M < 1.4 M sun b)Neutron Star – M > 1.4 M sun c)Black Hole – M > 3 M sun For a black hole the mass is compacted into a single point called a singularity. Ideally this point would have zero volume and therefore infinite density. Everything that enters a black hole (passes the event horizon) is pulled into the singularity, including light! R s – Schwartzchild radius Extremely dense star (R ~ 10 km) 1 teaspoon weighs 5 billion tons!

The black hole can emits high energy electromagnetic radiation, as shown in the illustration below. This electromagnetic radiation is not in the visible spectrum. A star loses mass to a black hole. This is the result of a single star of a binary pair collapsing into a black hole, or possibly a single star exploding and creating a black hole and another star. An astronaut entering a black hole. The gravitational forces are so strong that the gravitational force at the astronauts head is stronger than the gravitational force at his feet. This causes the stretching and eventually the astronaut would be torn apart.