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### Transcript of Chapter 13 Gravity - Austin Community · PDF fileChapter 13 Gravity. MFMcGraw-PHY 2425...

• Chapter 13

Gravity

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 2

Gravity

Keplers Laws

Newtons Law of Gravity

Gravitational Potential Energy

The Gravitational Field

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 3

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 m m

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 4

Characteristics of Gravity

1 2g 2

12

m mF

r

1 2g 2

12

m mF = G

r

g 2

1F

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

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 5

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.

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 6

Characteristics of Gravity

Gravity is a vector quantity

1 212 122

12

m mF = G r

r

12 21F = - F

12 21r = - r

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 7

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 Brahes data when he died.

Kepler deduced his three laws from this data

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 8

Keplers Laws

1. Law of Orbits - ellipse

2. Law of Areas - equal area in equal times

3. Law of Periods -

22 3

s

4T = r

GM

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 9

Keplers 1st Law

Perigee Apogee

(Closest approach) (Farthest approach)

Elliptical

shaped orbit

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 10

Construction of an Ellipse

r1 +r2 = 2a

a = semi major axis

b = semi minor axis

F are the focal points

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 11

Keplers 2nd Law

Equal areas are swept out in equal times

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 12

Derivation of Keplers 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

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 13

rava = rpvp

Conservation of Angular Momentum

L = mrvsin = constant, therefore rvsin is constant with r,

v and all changing but the product of the three remaining

constant in value

Evaluate the

expression at = 900

where sin = 1, at

apogee and perigee

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 14

Keplers LawsVariation of periods with mean orbital radius

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 15

Keplers 3rd Law

22 3

s

4T = r

GM

The Period Equation

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 16

Keplers 3rd Law

22 3

s

4T = 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.

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 17

The Determination of G

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 18

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

wasnt 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 Cavendishs old data.

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

might still be making discoveries after youre dead.

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 19

The Cavendish Experiment(minus the environmental shielding)

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 20

The Cavendish Experiment

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 21

Actual Cavendish Experiment Data

Each tiny square is 1 minute.

Scr

e en

Def

lect

ion

- 1

.5m

aw

ay

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 22

Gravitational Potential Energy

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 23

Bound State - Unbound State

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 24

EGM mU(r) = -r

Escape Speed of a Projectile

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 25

Gravitational Field

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 26

Preview of Electric Field Calc.-EPII

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 27

g-Field of a Thin Spherical Shell

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.

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 28

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

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 29

g-Field Distribution - Solid Sphere

3 34 43 3

3

3

3

2 2 3 3

M' M=

r R

rM' = M

R

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

r r R R

for r R

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 30

Homework Problems

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 31

Gravitational force on m and G field when m is gone

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 32

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

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 33

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

center under their mutual gravitational attraction?

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 34

Extra Slides

• Chap13Ha-Gravity-Revised 10/13/2012MFMcGraw-PHY 2425 35