# Gravitation Newton's Law of Gravitation; Kepler's Laws of ... Gravitation Newtons Law of Gravitation; Keplers Laws of Planetary Motion. Lecture 14 Monday: 1 March 2004 Physical Principles of Design Forces and Motion Learning to analyze and predict the motion of objects moving with constant accelerations.Topics of kinematics, projectile motion . Coming to understand the causes of motions and

changes in motion-topics of (linear and rotational) forces, momentum, energy Forces we have encountered:frictional force, normal force, applied forces of pushes/pulls, tension in strings, and the GRAVITATIONAL FORCE How Many Different Forces are There? All of the forces we have worked with or

will work with are specific examples of following general types of forces: The Fundamental Forces of Nature: 1. Gravitational Force 2. Electromagnetic Force 3. Strong Force (Nuclear) 4. Weak Force (Nuclear) A General Expression for the Gravitational Force More generally,

m1m2 F G 2 (This force is always Attractive) r G 6.67 10 11 2 Nm /kg

2 Revisiting Gravitational Force F mg More Specifically, FFrom Earth on an Object mobject g

Where , g is the acceleration due to the earths gravitational attraction. It is not only the earth that attracts other objects. Any object with mass will attract any other object with mass. At the surface of the Earth Mm F G 2 R where M is the mass of Earth,

m is the mass of the object R is the Radius of the Earth Mm But since F mg with F G 2 R GM g 2 (acceleration due to gravity on the earth) R

Characteristics of the Gravitational Force m1m2 F G 2 The force is always attractive. r There is a Newtons third law force pair involved. It acts along a line connecting the centers of the two objects (called a Central Force) It is inversely proportional to r2 (called a one over r squared force) Experimental measurement show us that it is a conservative

force (the gravitational force on earth is conservative-remember? This is a general expression of that same force) Defining the Potential Energy Associated with this Force U U b U a Wab b b a

a Wab Fds F dr m1m2 F G 2 r POTENTIAL ENERGY Choose U = 0 at r = r

U F dr GMm U r Gravitational Potential Energy Near Earth GMm U

R KEPLER'S LAWS 1. The Law of Orbits: All planets move in elliptical orbits having the Sun at one focus. 2. The Law of Areas: A line joining any planet to the Sun sweeps out equal areas in equal times. The Law of Areas A 21 (r )r

dA 1 2 d 1 2 2 r 2 r dt dt L constant L rmv rm r 2 L mr KEPLER'S LAWS

3. The Law of Periods: The square of the period of any planet about the Sun is proportional to the cube of the semimajor axis of its orbit. The Law of Periods F ma GMm 2 m r 2 r

2 GM 2 2 3 T r 2 ( 2

) T2 r3 GM ENERGY IN CIRCULAR ORBITS GM K mv m r GMm

K 2r GMm U r GMm E U K 2r 1 2 2

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