1

Universal gravitation

2

Let’s look at this pretty boy and this pretty girl …

3

Let’s look at this pretty boy and this pretty girl …

Do they exert a force on

one another?

4

Do they exert a force on one another?

They’re not in contact…But they ARE exerting a force on one another…What is that force?

5

Newton’s Law of Gravity

There is a force of attraction (GRAVITATIONAL FORCE) between each pair of objects in the universe that is proportional to the masses of the

objects inversely proportional to the distance

between them

6

Newton’s Law of Gravity

For two masses m1 and m2:

m1

m2

Fon 1, due to2Fon 2, due to 1

SAME MAGNITUDE, OPPOSITE DIRECTIONS!

7

Newton’s Law of Gravity

The magnitude of the gravitational force exerted by m2 on m1 is:

The direction of the force is always towards the other particle – the force is ATTRACTIVE!

1 22g

Gm mF

r

11 2 26.673 10 /G N m kg

8

Let’s go back to the pretty boy and the pretty girl

mB=72kg

mA=54kg

r=2m

9

What is the gravitational force exerted by: The pretty boy on the pretty girl? The pretty girl on the pretty boy?

10

We know that (by Newton’s 3rd Law), the MAGNITUDE of these forces are the SAME, but OPPOSITE in DIRECTION.

FA on B FB on A

11

The magnitude is given by

2

11 2 2

2

(6.673 10 / )(72 )(54 )

(2 )

A Bg

Gm mF

r

N m kg kg kg

m

86.486 10 N EXTREMELY SMALL!

12

Gravitational Force

So we see that for small masses, gravitational attraction is quite negligible (but it’s there).However, if at least one of the masses is large (like a planet, for example) gravitational attraction is LARGE!

13

Gravitational Force

For large masses, r is taken to be

the center-to-center distance. That is, we treat them like

PARTICLES

14

The Earth and the Moon

What is the magnitude of the gravitational force exerted by the Earth on the Moon? (mmoon=7.36 x 1022kg, mEarth=5.98 x 1024kg, center-to-center distance from Earth to moon = 3.84 x 108m)

2Earth Moon

gEarthtoMoon

Gm mF

r

15

The Earth and the Moon

2

11 2 2 24 22

8 2

(6.673 10 / )(5.98 10 )(7.36 10 )

(3.84 10 )

Earth Moong

Gm mF

r

N m kg kg kg

m

201.99 10 N LARGE!

16

Mass and weight

Mass and weight are two intrinsically different notions. The mass is a measurement of how much matter is in an object, while weight is a force, and measures how hard gravity is pulling on that object. A better scientific definition of mass describes it as having inertia, which is the resistance of an object to being accelerated when acted on by an external force. Your mass is the same wherever you are because the amount of stuff you're made of doesn't change, but your weight depends on how much gravity is acting on you at the moment;

17

Mass and weight

The mass is measured in kilos or (pounds) and the force is measured in Newton. A Newton is the force that would give a mass of one kilogram an acceleration of one meter per second per second.

18

Mass and weightA bathroom scale measures … The

weight or the mass?

19

Mass and weight

A bathroom scale measures … The weight or the mass?

THE WEIGHT!

Not the mass, because it measures the effect of the gravity over your mass.

20

The bathroom scale should give us a reading in Newton, rather than in kilos, but it is convenient to identify weight and mass in most applications.

21

Weight on Earth

Your weight is just the force exerted by the earth on you.What is the weight of an object with mass m on the surface of the Earth?

2Earth

gcenteroftheEarthtoYou

Gm mF

r 2

Earth

E

Gm m

R

6380ER km

22

Weight on Earth

But we know that your weight is just equal to

sogF mg

2Earth

E

Gm mmg

R

6380ER km

23

Weight a distance d from the surface of the Earth

What if mass m is a distance d from the surface of the Earth? (common scenario: airplanes, mountain climbing, etc.)Then the distance between the center of the Earth and the mass would change.Gravitational force on mass m, and hence, its weight will change!

24

Weight a distance d from the surface of the Earth

A mass m is a distance d away from the surface of the earth.

centerofEarthtoMassm Er R d

So the weight of mass m (=gravitational force exerted by earth on mass m) is

2( )Earth

E

Gm m

R d

25

Weight a distance d from the surface of the Earth

So the weight of mass m (=gravitational force exerted by earth on mass m) is

2( )Earth

E

Gm m

R d

SMALLER THAN THE

WEIGHT ON THE

SURFACE OF THE EARTH!

26

Weight a distance d from the surface of the Earth

mWeight varies inversely with the square of the distance from the earth’s center

2r

mGg E

r= RE

W(N)

700

500

300

0

200100

400

600m

W= mgo

r > RE

r (x 106 m)

27

Weight a distance d from the surface of the Earth

Problem: At what distance above the surface of the earth is the acceleration due to gravity 0.980 m/s2 if the acceleration due to gravity at the surface has a magnitude of 9.80 m/s2?

28

Weight a distance d from the surface of the Earth

From Newton’s Law of Gravity:

You can plug in the values of G, mEarth, RE, and solve for r’, but you also know that, at the surface of the Earth

22

0.980 / Earth

g

Gm mF m s

r

22

, 9.8 / Earthg

E

Gm mF mg g m s

R

(1)

(2)

29

Weight a distance d from the surface of the Earth

Divide (2) by (1):

EarthGm m2E

Earth

R

Gm m

2

2

2

9.8 /

0.98 /

mg m s

mg m s

r

(3)

(2)2 2

2 2

9.8 /

0.98 /E

r m s

R m s

30

Weight a distance d from the surface of the Earth

Solving for r’:

But r’ is the distance from the CENTER (NOT the SURFACE) of the Earth to mass m:

(3)

(2)

10 Er R

10E Er R d R

( 10 1) 14,000Ed R km

31

Weight ElsewhereThe mass of Venus is 4.87 x 10ˆ24 kg, and its radius is 6,051,000 m.

(a) What is the acceleration due to gravity on the surface of Venus?

(b) What is the weight of a 5.00 kg rock on the surface of Venus?

32

Weight Elsewhere (a) What is the acceleration due to gravity

on the surface of Venus?

2venus

Venusvenus

m mmg G

R

2, 88.8s

mg sVenus

2

2,

81.9 s

m

R

mGg

E

EsEarth

Plug in the data … and get

33

Weight Elsewhere (b) What is the weight of a 5.00 kg rock

on the surface of Venus?

Ns

mkgmgW sVenus 4.4488.800.5

2,

Ns

mkgmgW sEarth 1.4981.900.5

2,

34

Mass and weightSo, to summarize: Your mass is the same wherever you are, but your weight depends on how much gravity is acting on you at the moment.

35

Multiple ObjectsThree particles are arranged as shown. What is the net gravitational force that acts on particle A due to the other particles?

Particle Mass, kg

mA 6.0

mB 4.0

mC 4.0mA

mB

mC

4.0cm

2.0cmFBA

FCA

36

Multiple ObjectsSo what do we do with the forces????

We must add them as vectors!!!

Particle Mass, kg

mA 6.0

mB 4.0

mC 4.0

mA

mB

mC

4.0cm

2.0cmFBA

FCA

22, ACABAnet FFF

37

Multiple Objects

mA

mB

mC

4.0cm

2.0cmFBA

FCA

2ˆA B

B AAB

Gm mF j

r

2ˆ( )A C

C AAC

Gm mF i

r

22

2 2A CA B

onAAB AC

Gm mGm mF

r r

38

Multiple Objects

Plugging in the numerical values, we get: 6

2ˆ ˆ4.00 10A B

B AAB

Gm mF j N j

r

62

ˆ ˆ( ) 1.00 10A CC A

AC

Gm mF i N i

r

2 26 6 64.00 10 1.00 10 4.10 10onAF N N N

39

Multiple Objects

mA

mB

mC

4.0cm

2.0cmFBA

FCA

76 104

40

Summary

To apply Newton’s Law of Gravity: What are the masses involved? What is the distance between them?

(remember, we need the CENTER-TO-CENTER distance)

1 22g

Gm mF

r EVERYTHING’

S HERE!

41

Summary

The acceleration due to gravity near the surface of the Earth

2Earth

E

Gm mmg

R

42

How was the constant G measured? Click here to see how http://www.youtube.com/watch?v=4JGgYjJhGEE&feature=related

43

Henry Cavendish was born on 10 October 1731 in Nice, France, where his family was living at the time. The Cavendish experiment, performed in 1797–98 was the first experiment to measure the force of gravity between masses in the laboratory and the first to yield accurate values for the gravitational constant.[

44

The apparatus constructed by Cavendish was a torsion balance made of a six-foot wooden rod suspended from a wire, with a 2-inch diameter 1.61-pound (0.73 kg) lead sphere attached to each end. Two 12-inch 348-pound lead balls were located near the smaller balls, about 9 inches away, and held in place with a separate suspension system. The experiment measured the faint gravitational attraction between the small balls and the larger ones.

45

46

Planet Mass (kg) Radius (m)

Mercury 3.30 x 10ˆ23 2,440,000

Venus 4.87 x 10ˆ24 6,051,000

Earth 5.97 x 10ˆ24 6,378,000

Moon 7.35 x 10ˆ22 1,738,000

Mars 6.42 x 10ˆ23 3,397,000

Jupiter 1.90 x 10ˆ27 71,492,000

Saturn 5.69 x 10ˆ26 60,268,000

Uranus 8.66 x 10ˆ25 25,559,000

Neptune 1.03 x 10ˆ26 24,764,000

Pluto 1.31 x 10ˆ22 1,160,000

47