The moon is essentially gray, no color. It looks like plaster of Paris, like dirty beach sand with lots of footprints in it. -James A. Lovell (from the Apollo 13 mission) David J. Starling Penn State Hazleton Fall 2013
Objectives for Chapter 13 (a) Use the concept of gravitational field to analyze the motion of objects near the surface of the Earth and in planetary motion. (b) Analyze the orbital motions (including escape velocity) using the gravitational potential energy of the system.
A cannon fires a ball vertically upward from the Earth s surface. Which one of the following statements concerning the net force acting on the ball at the top of its trajectory is correct? (a) The net force on the ball is instantaneously equal to 0 newtons at the top of the flight path. (b) The direction of the net force on the ball changes from upward to downward. (c) The net force on the ball is less than the weight, but greater than zero newtons. (d) The net force on the ball is greater than the weight of the ball. (e) The net force on the ball is equal to the weight of the ball.
The gravitational force is a mutual force between two separated objects (distance r) of masses m 1 and m 2 given by F = G m 1m 2 r 2 G = 6.67 10 11 m 3 /kg-s 2.
If an object at the surface of the Earth has a weight W, what would be the weight of the object if it was transported to the surface of a planet that is one-sixth the mass of Earth and has a radius one third that of Earth? (a) 3W (b) 4W/3 (c) W (d) 3W/2 (e) W/3
From Newton s third law, we know that this force must have an equal but opposite pair; i.e., object 1 pulls on object 2, but object 2 also pulls on object 1.
Shell Theorem: a uniform sphere of matter attracts a particle that is outside as if all the sphere s mass were concentrated at its center. Uniform spherical objects just become points.
Principle of Superposition: If N objects interact with particle 1 gravitationally, the total force is just the vector sum. F 1,net = F 12 + F 13 + + F 1N F 1,net = N i=2 F 1i
Example 1: Three particles of masses m 1 = 6.0 kg and m 2 = m 3 = 4.0 kg interact gravitationally as shown below. If a = 2.0 cm, what is the net gravitational force on particle 1 from 2 and 3?
We can apply to an object (m) near the surface of the Earth (M): F = G Mm r 2 a g = GM r 2 = ma g
The force due to gravity on the surface of the earth is not consistently 9.83 m/s 2. Earth is not a perfect sphere; the mass within the Earth is not uniformly distributed; Earth rotates.
Example 2: Find the percent difference in weight (F N ) for a crate at the equator relative to a crate at the north pole.
Example 3: An astronaut of height 1.70 m floats feet down a distance r = 6.77 10 6 m away from the center of (a) Earth, (b) a black hole of mass 1.99 10 31 kg. Find the difference in the gravitational force between her head and her toes.
Gravity is a conservative force, so lets find its potential energy using U = W. U = W U( ) U(R) = R F(r) d r 1 = GMm R r 2 dr [ ] GMm = r R U(R) = 0 + GMm R U(R) = GMm R
The change gravitational potential energy U is path independent. U = W
The force from this potential energy is just the derivative (since we used an integral to derive it). F(r) = du dr = d ( GMm ) dr r = GMm r 2 The minus sign indicates the force points radially inward.
Example 4: You stand on the surface of a planet of radius R and mass M. Derive the speed at which a projectile would need to be launched to escape the pull of the planet s gravity. This is known as the escape speed.
A large asteroid collides with a planet of mass m orbiting a star of mass M at a distance r. As a result of the collision, the planet is knocked out of its orbit, such that it leaves the solar system. Which of the following expressions gives the minimum amount of energy that the planet must receive in the collision to be removed from the solar system? (a) GMm/r (b) GMm/r 2 (c) GMm/ r (d) Gm/r (e) Gm/r 2
Example 5: An asteroid is headed toward Earth. Its speed is measured to be 12 km/s when it is 10 earth radii from the center of the planet. What is its impact speed, ignoring drag from the atmosphere?
Johannes Kepler was a 17th century mathematician who developed three laws of planetary motion. 1. The Law of Orbits: all planets move in elliptical orbits with the Sun at one focus.
2. The Law of Areas: a line that connects a planet to the Sun sweeps out equals areas in equal time intervals (i.e., da/dt = constant). A = 1 2 r2 ( θ) da = 1 2 r2 dθ da = 1 dθ r2 dt 2 dt = 1 2 r2 ω L = rp = rmv = rmrω da = L dt 2m
3. The Law of Periods: the square of the period of any planet is proportional to the cube of the semimajor axis of its orbit. F = ma GMm r 2 = m(rω 2 ) = mr 2 ( 4π 2 T 2 = GM ) r 3 ( ) 2π 2 T
Example 6: Halley s comet orbits the sun with a period of 76 years and, in 1986, had a distance of closest approach to the Sun at 8.9 10 10 m. (a) What is its farthest distance from the Sun? (b) What is the eccentricity e of the orbit?
When one object orbits a much larger object, mechanical energy is conserved. For a circular orbit, F = ma GMm r 2 = m v2 r GMm = 1 2r 2 mv2 = K E = K + U = GMm GMm 2r r E = GMm 2r (for an elliptical orbit, E = GMm/2a)
The total energy of an orbiting body is negative. E = GMm 2r
Example 7: An astronaut releases a bowling ball of mass 7.20 kg into a circular orbit about Earth at an altitude of 350 km. (a) What is the mechanical energy of the ball in its orbit? (b) What is the mechanical energy of the ball on the surface of the Earth?