GravitationNewton’s Law of Gravitation;

Kepler’s Laws of Planetary Motion.

Lecture 14

Monday: 1 March 2004

Physical Principles of DesignForces 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 Force2. Electromagnetic Force3. Strong Force (Nuclear)4. Weak Force (Nuclear)

A General Expression for the Gravitational Force

2211

221

kg/mN1067.6

)Attractive always is force (This

generally, More

G

r

mmGF

Revisiting Gravitational Force

gmF

More Specifically,

gmF objectFrom

Objectan on Earth

Where , is the acceleration due to the earth’s gravitational attraction.

It is not only the earth that attracts other objects.

Any object with mass will attract any other object with mass.

g

At the surface ofthe Earth

earth) on thegravity todueon accelerati(

with sinceBut

Earth theof Radius theis R

object theof mass theis m

Earth, of mass theis M where

2

2

2

R

GMg

R

MmGFgmF

R

MmGF

Characteristics of the Gravitational Force

2

21

r

mmGF

•The force is always attractive.

•There is a Newton’s 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

b

a

b

aab

abab

drFdW

WUUU

sF

2

21

r

mmGF

POTENTIAL ENERGY

• Choose U = 0 at r =

U F dr

UGMm

r

r

Gravitational Potential Energy Near Earth

R

GMmU

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 r r

dAdt

rddt

r

L

L rmv rm r

L mr

12

12

2 12

2

2

( )

constant

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

rm r

GM

r T

TGM

r

22

32

2

22

3

2

2

( )

ENERGY IN CIRCULARORBITS

K mv mGM

r

KGMm

r

UGMm

r

E U KGMm

r

12

2 12

2

2