Ch. 12.pps

Ch. 12: Gravitation

Why is the Earth round?

Why do animals have bones?

Why do we live in a “solar system?”

Why is there a Moon?

(12.1) Newton’s Law of

Gravitation

•Note: 3rd Law tells us F

g(1 on 2) = - Fg(2 on 1)

→ →

1798: G found to be 6.673 x 10-11 N m2/kg2

F

=

G m

m

r

Find the magnitude and direction of the net

gravitational force exerted on the small star by the two large stars.

Note: A sphere can be treated as though all its mass is concentrated at its center

Inner Space

Down in a cave below the surface of the Earth there is

F

for Objects Near the Earth

M

m

R

r

(r > R_E)

F

for Objects On the Earth’s Surface

F

=

M

m

R

F

for Objects Inside the Earth

M

m

R

(r < R_E)

*Using calculus, it can be shown that when r < R_E

only the mass within r exerts a force!

r

M

But what is M?

F

r

“linear”

“inverse square”

Where is F_g largest/smallest?

R_E

Inner Space

Down in a cave below the surface of the Earth there is

Weight

•when an object is on the Earth, F_g from the rest of the universe may be ignored

F

=

G

M

m

R

•_{so the only gravitational force is:}

mg

G M

m

R

•_{we call this “weight” (a.k.a. mg), thus:}

g

G M

R

Example 12.3 Mission to Mars

R_M = 3.4 x 106 m M

M = 6.42 x 1023 kg

w_lander = 39,200 N

(12.3) Gravitational Potential

Energy

U

=

mgy

Where did this come from? F = − dU

dy = −mg F_g = −mg y



Guess:

U

= −

G m

m

r

Check:

−

dU

dr

= −

G m

m

r

F

=

−

G m

m

r

r



M

m

R

r

F

For objects at any distance from Earth:

U =

−

G M

m

Notes:

1. U always < 0

2. As r increases, U increases (less negative)

3. As r decreases, U decreases (more negative)

U

r

This is a bound system (recall marble analogy)

R_E

Problem 12.65

A hammer with mass

m

is dropped

from rest from a height

h

above the earth’s

surface. This height is not necessarily small

compared to the radius

R

of the earth. If

you ignore air resistance, derive an

expression for the speed

v

of the hammer

when it reaches the surface of the earth.

Your expression should involve

h

,

R

, and

Example 12.5 Escape Speed

M

m

R

v

Note: To escape means to barely reach r= as

you come to rest!

U

r

R_E

−G M_E m R_E

Example 12.5 Escape Speed

v_escape = 2 G M_E R_E

v_escape = 1.12 x 104 m/s  25,000 mi/h

Why launch from Cape Canaveral?

M

m

R

v

m

•_{escape from Earth}

is extreme!

(12.4) Motion of Satellites

•_{if launched at lower}

speeds, an object

Video Quiz

1. How does g_Earth compare to g_Moon?

2. Why are astronauts “weightless” in their spacecraft?

3. What is a satellite?

4. How far away are our closest satellites? Our farthest?

5. What are the periods of these satellites, respectively?

(12.4) Motion of Satellites

•_{Let’s consider circular orbits, to which Newton’s}

R

r

v

F_g

G M

m

r

=

mv

r

v = G M_E r

•_{for a given radius r, the speed of circular orbit}

is uniquely determined

•_{orbital speed is independent of satellite’s mass}

•_{since v = 2}r/T, we can find an expression for the period of a circular orbit

T = 2  r3/2

GM_E

•for orbits with larger r, speed is ______ and period is _______

Note:

slower longer

e.g. ISS Moon

r = 6800 km

T = 93 min v = 7.7 km/s

r = 384,000 km

T = 27 days v = 1 km/s

R_E r

v

M_E m

Notes:

1. for a given r, E is uniquely determined

3. as r increases, E increases (less negative)

4. As r decreases, E decreases (more negative)

Example 12.6

Suppose you want to place a 1000-kg weather satellite 300 km above the earth’s surface.

a) What speed and period must it have?

b) How much work must be done to place it in orbit?

c) How much additional work would have to be done to make this satellite escape the earth?

A space shuttle and a satellite are in the

same circular orbit, but on opposite sides

of the Earth. How can the

shuttle catch up to the

satellite?

(12.5) Kepler’s Laws of

Planetary Motion

1619: Kepler, building on earlier work of

Copernicus and Brahe, discovered three laws describing the motion of planets

(12.5) Kepler’s Laws of

Planetary Motion

1619: Kepler, building on earlier work of

Copernicus and Brahe, discovered three laws describing the motion of planets

2. A line from the sun to a planet

sweeps out equal areas in equal times.

This can be explained using conservation of angular momentum:

L = mvr

3. The periods of the planets are proportional to the 3/2 power of the semi-major axis of their

orbits.

T = 2



a

GM

For a satellite in circular orbit around the Earth, a = r and M_S  M_E