Download PHYS16 - Lecture 26

PHYS16 – Lecture 26
Gravitation
November 8, 2010
Review of Last Week…
Practice Question 1
• I want to use a wheelbarrow to move 30 kg of
yard waste with a center of mass 0.2 m from the
wheel axle. If the handle is located at 1 m from
the wheel axle how much force should I use to
keep the wheelbarrow lifted?
A)
B)
C)
D)
300 N
60 N
50 N
1500 N
F=(30kg)(9.8 m/s^2)(0.2 m)/(1 m) = 60 N
• What is my mechanical advantage?
MA= 5
Practice Question 2
• You have a meter stick that balances at the 50cm mark. Is it possible for your meter stick to
be inhomogeneous?
A) Yes
B) No
C) Don’t know
This Week
•
•
•
•
Newton’s law of Gravity
Gravitational Potential Energy
Satellites
Kepler’s Laws of Planetary Motion
– Orbital Shape
– Orbital Areas
– Orbital Period
Gravitational Force
Newton’s law of Gravity
• Gravity – an attractive force between two
masses
m1m2
FG  G 2
r
• G= gravitational constant = 6.674E-11 Nm2/kg2
• Acts at the center of mass
http://scienceblogs.com/startswithabang/upload/2009/06/the_last_100_years_1919_einste/300px-NewtonsLawOfUniversalGravitation.svg.png
Gravity – in the heavens and on earth
• The moon and a falling apple behave in the
same way…
m1m2
FG  G 2
r
FG ,Earth surface
M E m2
G
 m2 g
2
RE
• gapple=9.81 m/s2
• gmoon=0.00272 m/s2=gapple/3600
Disputing Gravity
Required given Newton’s system of
accelerations being enacted by forces
There is a force
of attraction
Proof: apples, moon, celestial bodies fall
towards each other
between all objects
Proof: 130 years later by Cavendish, but at
the time seemed nice not to distinguish
across empty space,
between an apple and a planet
proportional to m
Proof: None at the time. Galileo said there was no
dependence of gravity on mass. Later, Cavendish
and to M
experiment proves.
and to 1/r2.
Proof: Cavendish experiment. However, no way to
measure mass of the sun or planet independently.
Adapted from Physics for Poets by Robert March
Proof: Comparison of moon’s acceleration to that of an apple,
Kepler’s Laws
Disputing Gravity
There is a force
of attraction
between all objects
across empty space,
proportional to m
and to M
and to 1/r2.
Adapted from Physics for Poets by Robert March
So called “Spooky action at a distance.” Einstein
later shows that gravity leads to curvature in
space-time. Is there a gravitational particle? Is
gravity just a product of entropy?
Example Question: Gravity on Jupiter
• What is the weight of a 65 kg person on
Jupiter? (RJ=7.15E7 m, MJ=1.9E27 kg)
MJm
FJ  G 2
RJ
FJ  1610 N
Example Question: Mars and the Earth
• How big is the gravitational force between
Mars and the Earth?
(r=1.36E8 km, MM=6.42E23 kg, ME=5.97E24 kg)
MEMM
FG  G
r2
FG  1.38E16 N
So why doesn’t Mars revolve around the Earth?
Gravitational Potential
Gravitational Potential Energy
• Potential Energy associated with being in a
gravitational field
 
U    F  dx
r
Mm
U (r )  U ()   G 2 dr
r

Mm
U (r )  U ()  G
0
r
Mm
U (r )  G
r
Satellites
Weightlessness
• Why do astronauts feel weightless? Isn’t there
still a force of gravity on them?
Yes, just not a normal force!
So why doesn’t Moon fall into earth?
Centripetal Force and Gravity
• For an object (like a satellite) in circular
motion due to gravity
Fc  FG
Fc = FG
http://qwickstep.com/search/earth-orbit-around-the-sun.html
Mm
mac  G 2
r
GM
2
v 
r
Example Question: Moon Energy
• If the Moon-Earth distance were to shrink
what would happen to the Moon’s kinetic
energy?
A) Increase
B) Stay the same
C) Decrease
Example Question: Moon Period
• If the Moon-Earth distance were to shrink
what would happen to the period of the
moon?
A) Increase (Greater than ~28 days)
B) Stay the same at ~28 days
C) Decrease (Less than ~28 days)
Orbital Energy
• Gravitational potential energy increases as
distance increases
• Kinetic energy decreases as distance increases
• So, why do higher orbitals have more energy?
E  K U
1 2 GMm GMm GMm
E  mv 


2
r
2r
r
GMm
E
2r
Example Question: Satellite
• A Satellite orbiting the Earth wants to go to a
lower orbit. What should the satellite do?
A) Nothing. It is falling toward the earth.
B) Turn on rocket thrusters to accelerate and increase
speed, then move to lower orbital.
C) Turn on rocket thrusters to decelerate and decrease
speed, then move to lower orbital
Escape velocity
• Velocity needed to “escape” the gravitational
force
E  ( K f  K i )  (U f  U i )  0
1 2
GMm
(0  mvi )  (0 
)0
2
R
2GM
vi 
R
Doesn’t depend on mass of object, only depends
on the gravitational field…
Main Points
• Gravitational Force = GMm/r2
• Centripetal force = gravitational force for
object in orbit
• Gravitational Potential = GMm/r
– To get further from a massive object requires
more energy
– Gravitational potential = zero at infinity