Yet to an observer (m) outside the Earth, its mass (m E ) acts as if concentrated at the center
Newton’s Law of Gravitation This is the magnitude of the force that: m 1 exerts on m 2 m 2 exerts on m 1 What if other particles are present?
Superposition Principle the gravitational force is a vector so the gravitational force on a body m due to other bodies m 1, m 2,... is the vector sum: Do Exercise 12-8 Do Exercise 12-6 Do Exercise 12-8 Do Exercise 12-6
Superposition Principle Example 12-3 The total gravitational force on the mass at O is the vector sum: Do some of Example 12-3 and introduce Extra Credit Problem 12-42
Gravitational Potential Energy, U This follows from: Derive U = - G m 1 m 2 /r
Gravitational Potential Energy, U Alternatively: a radial conservative force has a potential energy U given by F = – dU/dr
Gravitational Potential Energy, U U is shared between both m 1 and m 2 We can’t divide up U between them Example: Find U for the Earth-moon system
Superposition Principle for U For many particles, U = total shared potential energy of the system U = sum of potential energies of all pairs Write out U for this example
Total Energy, E If gravity is force is the only force acting, the total energy E is conserved For two particles,
Application: Escape Speed projectile: m Earth: m E Find the speed that m needs to escape from the Earth’s surface Derive the escape speed: Example 12-5
Einstein’s Special Relativity all inertial observers measure the same value c = 3.0×10 8 m/s 2 for the speed of light nothing can travel faster than light ‘special’ means ‘not general’: spacetime (= space + time) is flat
Einstein’s General Relativity nothing can travel faster than light but spacetime is curved, not flat matter curves spacetime if the matter is dense enough, then a ‘black hole’ forms
If mass M is compressed enough, it falls inside its Schwarzschild radius, R s This curves spacetime so much that a black hole forms
Black Holes Outside a black hole, v and r for circular orbits still obey the Newtonian relationship: Also: from far away, a black hole obeys Newtonian gravity for a mass M
Black Holes Spacetime is so curved, anything that falls into the hole cannot escape, not even light Light emitted from outside the hole loses energy (‘redshifts’) since it must do work against the extremely strong gravity So how could we detect a black hole?