Lecture 16 Gravitation
Today s Topics: The Gravitational Force Satellites in Circular Orbits Apparent Weightlessness lliptical Orbits and angular momentum Kepler s Laws of Orbital Motion Gravitational Potential nergy
Newton s Law of Gravitation very particle exerts an attractive force on every other particle. A particle is a piece of matter, small enough in size to be regarded as a mathematical point. The force that each exerts on the other is directed along the line joining the particles. G -11 6.673 10 N m kg
Gravitational Attraction of Spherical Bodies The force is the same as if all the mass of the sphere were concentrated at its center.
ACT: arth and Moon I Which is stronger, arth s pull on the Moon, or the Moon s pull on arth? a) the arth pulls harder on the Moon b) the Moon pulls harder on the arth c) they pull on each other equally d) there is no force between the arth and the Moon e) it depends upon where the Moon is in its orbit at that time By Newton s Third Law, the forces are equal and opposite.
ACT: arth and Moon II If the distance to the Moon were doubled, then the force of attraction between arth and the Moon would be: a) one quarter b) one half c) the same d) two times e) four times The gravitational force depends inversely on the distance squared. So if you increase the distance by a factor of, the force will decrease by a factor of 4. F G Mm R
Force Vectors A planet of mass m is a distance d from arth. Another planet of mass m is a distance d from arth. Which force vector best represents the direction of the total gravitation force on arth? d a b c d e d m The force of gravity on the arth due to m is greater than the force due to m, which means that the force component pointing down in the figure is greater than the component pointing to the right. m F m GM (m) / (d) GMm / d F m GM m / d GMm / d 1
Defining weight The weight of an object on or above the earth is the gravitational force that the earth exerts on the object. The weight always acts downwards, toward the center of the earth. On or above another astronomical body, the weight is the gravitational force exerted on the object by that body. W G M R m W mg g G M R
How does g change with altitude? g G M r G M ( R + h ) g 0!
Circular Orbits Imagine a satellite orbiting around the earth at a distance r from the center F c G mm r m v r v GM r
One r, one v à One T GM p r v r T T p r 3 GM Determine the speed of the Hubble Space Telescope orbiting at a height of 598 km above the earth s surface. v ( 6.67 10 11 N m kg ) 5.98 10 4 kg 6.38 10 6 m + 0.60 10 6 m ( ) 7.56 10 3 m s ( 16900 mi h)
Geosynchronous Orbit T p R 3 GM Let T 4 hours R 4 10 6 meters from the center of the earth OR 36,000 km (,000 miles) from the surface
In the Space Shuttle Astronauts in the space shuttle float because: a) they are so far from arth that arth s gravity doesn t act any more b) gravity s force pulling them inward is cancelled by the centripetal force pushing them outward c) while gravity is trying to pull them inward, they are trying to continue on a straight-line path d) their weight is reduced in space so the force of gravity is much weaker Astronauts in the space shuttle float because they are in free fall around arth, just like a satellite or the Moon. Again, it is gravity that provides the centripetal force that keeps them in circular motion.
Apparent Weightlessness F c G mm r m v r
Conservation of Angular Momentum å F xt ma m Dv Dt Dp Dt The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero. I w 0 0 I F w F
Conservation of Angular Momentum and lliptical Orbits The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero. The angular momentum of a system remains constant (is conserved) if the net external torque acting on the system is zero. I w 0 0 I F w F
What does conservation of angular momentum have to do with satellites? Once in orbit, the only external force acting on the satellite is the gravitational force
Real Satellite Orbits Are generally elliptical! I A w A I P w P I mr w v mr A v r A A mr P r v r P P
Kepler s Laws 1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse.. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time. 3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/ power. (see slide 8)
Gravitational potential energy of an object of mass m a distance r from the arth s center:
Gravitational Potential nergy
How do we reconcile: with ΔU refers to a change in potential energy U 0 is referenced to the infinite separation between the masses That U < 0 denotes the fact that the gravitational force is attractive
Total mechanical energy of an object of mass m a distance r from the center of the arth:
scape from planet earth f f i i f i U K U K + + To just get away from the earth, v 0 as r 11. km/s 0 1 0» - e e f i R M G v R mm G mv