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) 300 NB) 60 NC) 50 ND) 1500 N

• What is my mechanical advantage?

F=(30kg)(9.8 m/s^2)(0.2 m)/(1 m) = 60 N

MA= 5

Practice Question 2

• You have a meter stick that balances at the 50-cm mark. Is it possible for your meter stick to be inhomogeneous?A) YesB) NoC) 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

• G= gravitational constant = 6.674E-11 Nm2/kg2

• Acts at the center of mass

221

r

mmGFG

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…

• gapple=9.81 m/s2

• gmoon=0.00272 m/s2=gapple/3600

gmR

mMGF

r

mmGF

E

EG

G

222

surfaceEarth ,

221

Disputing Gravity

There is a forceof attractionbetween all objectsacross empty space,proportional to mand to Mand to 1/r2.

Required given Newton’s system of accelerations being enacted by forces

Adapted from Physics for Poets by Robert March

Proof: apples, moon, celestial bodies falltowards each other

Proof: 130 years later by Cavendish, but atthe time seemed nice not to distinguish between an apple and a planet

Proof: None at the time. Galileo said there was nodependence of gravity on mass. Later, Cavendish experiment proves.

Proof: Comparison of moon’s acceleration to that of an apple,Kepler’s Laws

Proof: Cavendish experiment. However, no way tomeasure mass of the sun or planet independently.

Disputing Gravity

There is a forceof attractionbetween all objectsacross empty space,proportional to mand to Mand to 1/r2.

Adapted from Physics for Poets by Robert March

So called “Spooky action at a distance.” Einsteinlater shows that gravity leads to curvature inspace-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)

N 6101

2

J

J

JJ

F

R

mMGF

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)

N 1.38E16

2

G

MEG

Fr

MMGF

So why doesn’t Mars revolve around the Earth?

Gravitational Potential

Gravitational Potential Energy

• Potential Energy associated with being in a gravitational field

r

MmGrU

r

MmGUrU

drr

MmGUrU

xdFU

r

)(

0)()(

)()(2

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

http://qwickstep.com/search/earth-orbit-around-the-sun.html

r

GMv

r

MmGma

FF

c

Gc

2

2

Example Question: Moon Energy

• If the Moon-Earth distance were to shrink what would happen to the Moon’s kinetic energy?

A) IncreaseB) Stay the sameC) 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 daysC) 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?

r

GMmE

r

GMm

r

GMm

r

GMmmvE

UKE

2

22

1 2

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

R

GMv

R

GMmmv

UUKKE

i

i

ifif

2

0)0()2

10(

0)()(

2

Doesn’t depend on mass of object, only dependson 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