Gravity and Orbits Objectives Clarify a number of basic concepts Speed vs. velocity Acceleration, and its relation to force Momentum and angular momentum Gravity Understand its basic workings Understand how it leads to orbital motion Introduce escape velocity
Some basic definitions Speed Distance traveled divided by the time taken. Velocity More informative Speed + Direction Acceleration A change in velocity Force Most simply thought of as a push or a pull A force causes an acceleration
30 mi/hr 30 mi/hr
Question: Which of the following does NOT represent an acceleration? a) A planet orbiting around the Sun. b) A car traveling at constant speed around a curve. c) A car traveling at constant speed in a straight line downhill. d) A car slowing to a stop from 25 mph, in a straight line.
Newton s s First Law of Motion Every object remains either at rest, or moving in a straight line at constant speed, unless it is acted upon by a net force. What does this mean? No net force = no acceleration
Newton s s Second Law of Motion When a net force acts on a body, the acceleration: Is in the same direction as the force. Is equal to the magnitude of the force divided by the mass of the body being accelerated. a = F / m
Newton s s Third Law of Motion For every force ( action ), there is an equal and opposite reaction force. Forces come in pairs. The action and reaction forces act upon different objects. This is why we can walk.
A quick check Does this make sense to you? You are sitting in your seats. Are you moving? Does this mean that no forces are acting upon you?
Newton s s Law of Gravity Every object in the universe exerts an attractive force upon every other object in the universe. The force between any two objects: Is proportional to the product of the masses of the two objects. Decreases with the square of the distance between them. F G = GM 1 M 2 R 2
Question: If the Sun s gravity were to suddenly turn off: a) The Earth would move in a perfect circle around the Sun. b) Earth would move in a straight line at constant speed, in whichever direction it happened to be headed at that instant. c) Earth would move at a constant speed in a straight line, directly away from the Sun. d) Earth would move away from the Sun in a spiral motion.
Question: A black hole is an object on whose surface the force of gravity is so strong that not even light can escape. What would have to happen to the Sun in order for it to become a black hole? a) It would have to shrink in size. b) It would have to expand in size. c) It would have to decrease in mass. d) None--Newton s law of gravity states that the Sun can never become a black hole.
So, how does gravity lead to orbital motion?
Gravity, combined with sideways motion, leads to an endless falling motion, aka orbits Think of a satellite orbiting the Earth. Gravitational force always acts to pull the satellite towards the center of the Earth. At the same time, though, the satellite has a sideways motion. If the sideways motion is large enough, the satellite can always stay the same distance from the Earth, and always be falling towards it! v g
Momentum; another conserved quantity Momentum is the product of mass and velocity. It has directionality Like energy, it can be transferred between objects. The total amount present cannot, however, be changed.
Angular momentum; yet another conserved quantity Angular momentum is a rotational analogue of momentum. It is the product of the mass of a rotating (or orbiting) object, times its rotational speed, times the distance away from the center of rotation. In order to conserve angular rotation, an orbiting satellite, for example, must speed up as it approaches the Earth.
Question: In order to launch a satellite to Venus, we would: a) Launch in the direction of Earth s orbital motion, to get a boost. b) Launch opposite to the direction of Earth s motion, to remove some speed. c) Launch when the rocket carrying the satellite was aimed straight at Venus. d) Launch the satellite away from Venus, so as to put the it into an elliptical orbit.
Escape velocity This is the speed at which something must travel in order to escape from the gravity of an object (such as Earth) We actually derive it from energies--set the initial kinetic energy of the object equal to its gravitational potential energy. For Earth, M=5.97x10 24 kg, and R=6400 km. v esc = 2GM R = 2 6.67 10 11 5.97 10 24 6,400,000 =1.1 10 4 m/s =11 km/s
In reverse, the escape velocity is also a typical collision speed If an object starts from rest, a long distance from Earth, and then falls to Earth It starts with no kinetic energy As it falls, gravitational potential energy is converted to kinetic energy When it hits Earth, its kinetic energy equals the potential energy that it lost while falling It hits at escape velocity!