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Chapter 13

Gravity

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Gravity

Kepler’s Laws

Newton’s Law of Gravity

Gravitational Potential Energy

The Gravitational Field

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Characteristics of Gravity

Acts along a line connecting the center of mass of the two

bodies.

It is a central force which implies that it is a conservative

force.

It is only attractive. There are no repulsive gravitational

forces, i.e. there is no anti-gravity.

Its magnitude is proportional to the two masses

g 1 2F α m m

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Characteristics of Gravity

1 2g 2

12

m mF α

r

1 2g 2

12

m mF = G

r

g 2

1F α

r

The strength of the gravitational force decreases with

distance

Therefore

The value of G was determined from the data of

Henry Cavendish: G = 6.67x10-11 Nm2/kg2

The gravitational force is

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Characteristics of Gravity

Gravity is the weakest of all the forces. It takes large

masses to produce a gravitational force that is

noticeable.

Gravity acts most on the grand astronomical scale

where large masses are found.

However, it is the exact balance of the positive and

negative charges that effectively cancels the long range

electric fields that allows gravity to be observed even at

these large astronomical distances.

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Characteristics of Gravity

Gravity is a vector quantity

ˆ�

1 212 122

12

m mF = G r

r

� �

12 21F = - F

ˆ ˆ12 21r = - r

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Early Astronomers

Tycho Brahe (1546-1601)

• He gathered observational data over a twenty year

period of measuring planet positions.

Johannes Kepler (1571-1630)

• Kepler worked with Brahe

• He inherited Brahe’s data when he died.

• Kepler deduced his three laws from this data

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Kepler’s Laws

1. Law of Orbits - ellipse

2. Law of Areas - equal area in equal times

3. Law of Periods -

22 3

s

4πT = r

GM

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Kepler’s 1st Law

Perigee Apogee

(Closest approach) (Farthest approach)

Elliptical

shaped orbit

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Construction of an Ellipse

r1 +r2 = 2a

a = semi major axis

b = semi minor axis

F are the focal points

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Kepler’s 2nd Law

Equal areas are swept out in equal times

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Derivation of Kepler’s 2nd Law

� �

� �

12dA = r × vdt

r × mvdA = dt

2m

dA L=

dt 2m

F is a central force so L

is conserved and

therefore

dA= constant

dt

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rava = rpvp

Conservation of Angular Momentum

L = mrvsinφ = constant, therefore rvsinφ is constant with r,

v and φ all changing but the product of the three remaining

constant in value

Evaluate the

expression at φ = 900

where sinφ = 1, at

apogee and perigee

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Kepler’s LawsVariation of periods with mean orbital radius

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Kepler’s 3rd Law

22 3

s

4πT = r

GM

The Period Equation

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Kepler’s 3rd Law

22 3

s

4πT = r

GM

Astronomical Unit = Mean radius of earth orbit around sun

1 AU = 1.50 x 10-11 m = 93.0 x 106 mi

For orbits around the Sun 2 3

T (Years) = R (AU)

For other applications replace Ms in the equation above

with the mass of the body that the object is orbiting

around.

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The Determination of G

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The Determination of G

According to legend, i.e. one text book author copying the

historical aspects of physics from the authors that came before

him, Henry Cavendish made the first accurate measurement of

G in 1745.

It turns out that the legend is incorrect. In fact, Cavendish

wasn’t even interested in measuring the value of G. Henry

Cavendish wanted to measure the density of the earth.

75 years later another researcher was able to squeeze a value of

G out of Cavendish’s old data.

Moral of the story - take good notes while you’re alive and you

might still be making discoveries after you’re dead.

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The Cavendish Experiment(minus the environmental shielding)

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The Cavendish Experiment

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Actual Cavendish Experiment Data

Each tiny square is 1 minute.

Scr

e en

Def

lect

ion

- 1

.5m

aw

ay

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Gravitational Potential Energy

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Bound State - Unbound State

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EGM mU(r) = -

r

Escape Speed of a Projectile

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Gravitational Field

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Preview of Electric Field Calc.-EPII

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g-Field of a Thin Spherical Shell

ˆ�

�

2

GMg = - r; r > R

r

g = 0; r < R

g outside the mass distribution

is the same as if all the mass

was concentrated at the CM.

Requires uniform mass

distribution or a mass

distribution that only varies

with r.

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Gravitational Field Inside a Hollow Sphere

�

gF = 02

1 1 1

2

2 2 2

1 2

2 2

1 2

1

2

1

m A r= =

m A r

m m=

r r

Gmg =

r

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g-Field Distribution - Solid Sphere

≤

3 34 43 3

3

3

3

2 2 3 3

M' M=

πr πR

rM' = M

R

GM' GM r GMg(r) = - = - = - r

r r R R

for r R

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Homework Problems

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Gravitational force on m and G field when m is gone

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Force on mass m at distances: 3a, 1.9a and 0.9 a

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What must their speed be if they are to orbit their common

center under their mutual gravitational attraction?

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Extra Slides

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