Isaac Newton’s 3 rd Law and the Universal Law of Gravitation.

Isaac Newton’s 3rd Law and the Universal Law of

Gravitation

= a push or pull acting on an object.

Examples:

gravity = pull

electrostatic attraction = pull

electrostatic repulsion = push

Force

Newton’s Second Law of Motion:

The acceleration of an object is directly proportional to the force acting on it,

and inversely proportional to its mass.

In mathematical form:

Or alternatively: amF m

Fa

Example of Newton’s Second Law:

A package of cookies has mass m = 0.454 kilograms,

And experiences gravitational acceleration g = 9.8 meters/second2

How large is the force acting on the cookies?

pound) 1 Newtons (4.4

Newtons 4.4m/sec kg 4.4

)m/sec kg)(9.8 454.0(2

2

F

F

amF

Newton’s Third Law of Motion:For every action, there is

an equal and opposite reaction.

Whenever A exerts a force on B, B exerts a force on A that’s equal in size and opposite in direction.

All forces come in pairs.

Example of Newton’s Third Law:

Cookies push on hand: F = 1 pound, downward.

Hand pushes on cookies: F = 1 pound, upward.

Remove hand!

Earth pulls on cookies: F = 1 pound, downward.

Cookies pull on earth: F = 1 pound, upward.

THIRD Law states: force on Earth = force on cookies

SECOND Law states: acceleration = force divided by mass

Mass of Earth = 1025 x mass of cookies

Therefore, acceleration of cookies = 1025 x acceleration of Earth.

(Cookies reach a high speed while the Earth hardly budges.)

But…why do the cookies and the Earth exert a force on each other?

Newton’s Law of Gravity states that gravity is an attractive force acting between ALL pairs of massive objects.

Gravity depends on:

(1) MASSES of the two objects,

(2) DISTANCES between the objects.

Newton’s question: can GRAVITY be the force keeping the Moon in its orbit?

Newton’s approximation: Moon is on a circular orbit.

Even if its orbit were perfectly circular, the Moon would still be accelerated.

The Moon’s orbital speed:

radius of orbit: r = 3.8 x 108 m

circumference of orbit: 2pr = ???? m

orbital period: T = 27.3 days = ???? sec

orbital speed:

v = (2pr)/T = ??? m/sec = ? km/sec!

The Moon’s orbital speed:

radius of orbit: r = 3.8 x 108 m

circumference of orbit: 2pr = 2.4 x 109 m

orbital period: T = 27.3 days = 2.4 x 106 sec

orbital speed:

v = (2pr)/T = 103 m/sec = 1 km/sec!

Acceleration required to keep Moon on a circular orbit

28

232

8

3

2

m/sec 00272.0m 103.8

m) 10(

m 108.3

m/sec 10

:Moon For the

radius orbitalr speed, orbitalv

:ison accelerati required The

r

va

r

v

r

va

Ratio of Accelerations to Distances

????? m/sec 00272.0

m/sec 8.9

)m/sec 00272.0(aMoon theoforbit At the

m/sec 8.9

Earth) of radius(Earth theof surface At the

2

2

2

2

a

r

Bottom Line

If gravity goes as one over the square of the distance,

Then it provides the right acceleration to keep the Moon on its orbit (“to keep it falling”).

Triumph for Newton!!

Fig. 5-2, p.81

Fig. 5-3, p.82

Fig. 5-1, p.80

Location

Distance fromEarth's center (m)

Value of gm/s2

Earth's surface

6.38 x 106 m

9.8

1000 km above

surface

7.38 x 106 m

7.33

2000 km above

surface

8.38 x 106 m

5.68

3000 km above

surface

9.38 x 106 m

4.53

4000 km above

surface

1.04 x 107 m

3.70

5000 km above

surface

1.14 x 107 m

3.08

6000 km above

surface

1.24 x 107 m

2.60

7000 km above

surface

1.34 x 107 m

2.23

8000 km above

surface

1.44 x 107 m

1.93

9000 km above

surface

1.54 x 107 m

1.69

10000 km above

surface

1.64 x 107 m

1.49

50000 km above

surface

5.64 x 107 m

0.13

p.83

(4) Newton’s Law of Gravity:

The gravitational force between two objects

F = gravitational force

M = mass of one object

m = mass of the second object

r = distance between centers of objects

G = “universal constant of gravitation”

2r

MmGF

Example: What is gravitational force between Earth and

cookies?

:in numbers theplug sLet'

2211

6

24

2

kg/m Newtons 107.6

m 106.4Earth of radius

kg 454.0 cookies of mass

kg100.6 Earth of mass

G

r

m

M

r

MmGF

Example: What is gravitational force between Earth and

cookies?

pound 1 Newtons 4.4F

kg/m Newtons 107.6

m 106.4Earth of radius

kg 454.0 cookies of mass

kg100.6 Earth of mass

:in numbers theplug sLet'

2211

6

24

2

G

r

m

M

r

MmGF

Table p.85

p.85

p.89

p.90

Fig. Q5-3, p.92

Fig. Q5-19, p.93

Fig. Q5-25, p.93

Discovered:

three laws of motion,

one law of universal gravitation.

Isaac Newton (1643-1727): English