Physics 151: Lecture 27, Pg 1 Physics 151: Lecture 27 Today’s Agenda l Today’s Topic çGravity...

Physics 151: Lecture 27, Pg 1

Physics 151: Lecture 27 Physics 151: Lecture 27 Today’s AgendaToday’s Agenda

Today’s TopicGravityPlanetary motion


New Topic - GravityNew Topic - Gravity Sir Isaac developed his laws of motion largely to explain observations

that had already been made of planetary motion.

See text: 14

Sun

Earth

Moon

Note : Not to scale


GravitationGravitation(Courtesy of Newton)(Courtesy of Newton)

Things Newton Knew,1. The moon rotated about the earth with a

period of ~28 days.2. Uniform circular motion says, a = 2R4. Acceleration due to gravity at the surface of

the earth is

g ~ 10 m/s2

5. RE = 6.37 x 106

6. REM = 3.8 x 108 m

See text: 14.1



Things Newton Figured out,1. The same thing that causes an apple to fall from a tree to the ground is what causes the moon to circle around the earth rather than fly off into space. (i.e. the force accelerating the apple provides centripetal force for the moon)

2. Second Law, F = ma

So, acceleration of the apple (g) should have some relation to the centripetal acceleration of the moon (v2/REM).

See text: 14.1


Moon rotating about the Earth : Moon rotating about the Earth :

So = 2.66 x 10-6 s-1.

Now calculate the acceleration. a = 2R = 0.00272 m/s2 = .000278 g

1

27 3

1

864002 2 66 10 6

..

rot

dayx

day

sx

rad

rotx s-1

Calculate angular velocity :

= v / REM = 2 REM / T REM = 2 / T

=



Newton found that amoon / g = .000278 and noticed that RE

2 / R2 = .000273

This inspired him to propose the Universal Law of Gravitation:Universal Law of Gravitation:

|FMm |= GMm / R2

R RE

amoong

G = 6.67 x 10 -11 m3 kg-1 s-2

See text: 14.1


Gravity...Gravity...

The magnitude of the gravitational force FF12 exerted on an object having mass m1 by another object having mass m2 a distance R12 away is:

The direction of FF12 is attractive, and lies along the line connecting the centers of the masses.

212

2112

R

mmGF

R12

m1 m2FF12 FF21

See text: 14.1



Compact objects:R12 measures distance between objects

Extended objects:R12 measures distance between centers

R12

R12


Gravity...Gravity... Near the earth’s surface:

R12 = RE

» Won’t change much if we stay near the earth's surface.

» i.e. since RE >> h, RE + h ~ RE.

RE

m

M

h 2E

Eg

R

mMGF

FFg

See text: 14.1



Near the earth’s surface...

22E

E

E

Eg

R

MGm

R

mMGF

So |Fg| = mg = ma

a = g

All objects accelerate with acceleration g, regardless of their mass!

22 /81.9 smR

MGg

E

E Where:

=g

See text: 14.3


Example gravity problem:Example gravity problem:

What is the force of gravity exerted by the earth on a typical physics student?

Typical student mass m = 55kgg = 9.8 m/s2.Fg = mg = (55 kg)x(9.8 m/s2 )

Fg = 539 NFFg The force that gravity exerts on any object is

called its Weight

W = 539 N


Lecture 27, Lecture 27, Act 1Act 1Force and accelerationForce and acceleration

Suppose you are standing on a bathroom scale in Physics 203 and it says that your weight is W. What will the same scale say your weight is on the surface of the mysterious Planet X ?

You are told that RX ~ 20 REarth and MX ~ 300 MEarth.

(a)(a) 00.75.75 W (b)(b) 1.5 W

(c)(c) 2.25 W E

X


Lecture 27, Lecture 27, Act 1Act 1SolutionSolution

The gravitational force on a person of mass m by another object (for instance a planet) having mass M is given by:

2R

MmGF

E

X

E

X

FF

WW Ratio of weights = ratio of forces:

2

2

E

E

X

X

R

mMG

R

mMG

2

X

E

E

X

RR

MM

75.201

3002

E

X

WW (A)


Kepler’s LawsKepler’s Laws

Much of Sir Isaac’s motivation to deduce the laws of gravity was to explain Kepler’s laws of the motions of the planets about our sun.

Ptolemy, a Greek in Roman times, famously described a model that said all planets and stars orbit about the earth. This was believed for a long time.

Copernicus (1543) said no, the planets orbit in circles about the sun.

Brahe (~1600) measured the motions of all of the planets and 777 stars (ouch !)

Kepler, his student, tried to organize all of this. He came up with his famous three laws of planetary motion.

See text: 14.3


Kepler’s LawsKepler’s Laws

1st All planets move in elliptical orbits with the sun at one focal point.

2nd The radius vector drawn from the sun to a planet sweeps out equal areas in equal times.

3rd The square of the orbital period of any planet is proportional to the cube of the semimajor

axis of the elliptical orbit.

It was later shown that all three of these laws are a result of Newton’s laws of gravity and motion.

See text: 14.4


Kepler’s Third LawKepler’s Third Law

Let’s start with Newton’s law of gravity and take the special case of a circular orbit. This is pretty good for most planets.

See text: 14.4

R

vm

R

mMGF pps

2

2

R

TRm

R

mMG pps

2

2

)/2(

32

2 4R

GMT

s


Kepler’s Second LawKepler’s Second Law

This one is really a statement of conservation of angular momentum.

See text: 14.4

rR

mMGrRFR ps ˆˆ 2

0ˆˆ rrR

mMG ps

Constant vRmpRL P



See text: 14.4

Constant vRmpRL P

2. The radius vector drawn from the sun to a planet sweeps out equal areas in equal times.

ConstantdtdA

R

dR

dA



See text: 14.4

Constant vRmpRL P

R

dR

dA

dtML

dtvRRdRdA21

21

21

Constant2

ML

dtdA


Energy of Planetary MotionEnergy of Planetary Motion

A planet, or a satellite, in orbit has some energy associated with that motion.

Let’s consider the potential energy due to gravity in general.

See text: 14.7

F GMsmp

R2

W F(r)drr1

r2

GMsmp

r2r1

r2

dr

U U f Ui W GMsmp (1

rf

1

ri)

r

mGMU psDefine ri as infinity

U

r

U 1

r

RE

0


Energy of a SatelliteEnergy of a Satellite

A planet, or a satellite, also has kinetic energy.

See text: 14.7

rmv

mar

mGM ps2

2

We can solve for v using Newton’s Laws,

r

mGMmvUKE ps 2

21

r

mGM

r

mGM

r

mGME pspsps

22

Plugging in and solving,


Recap of today’s lectureRecap of today’s lecture

Chapter 13 Gravity Planetary motion

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Physics 151: Lecture 27, Pg 1 Physics 151: Lecture 27 Today’s Agenda l Today’s Topic çGravity...

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