PHYSICS 3 (GENERAL PHYSICS I)

Centripetal Force and Newtons Law of Gravitation

Centripetal Force

CENTRIPETAL FORCE

Centripetal Force is a force that tends to keep object

in moving around a circular arc or path

The magnitude of the centripetal force is the product

of an objects mass and its centripetal acceleration as

it moves around the circular path

The direction of the centripetal force is always

directed towards the center of the circle.

RECALL: Centripetal acceleration

An object traveling in a

circle, even though it moves

with a constant speed, will

have an acceleration (since

velocity changes direction)

This acceleration is called

centripetal (center-

seeking).

The acceleration is directed

toward the center of the

circle of motion

Centripetal acceleration and the angular

velocity

The angular velocity and the linear

velocity are related (v = r)

The centripetal acceleration can

also be related to the angular

velocity

t

r

r

va

t

v

rr

vv

r

r

v

v

a

since,

Thus: rar

va CC

22

or

Similar triangles

Lengths of the sidesr

r

v

v

Forces Causing Centripetal Acceleration

Newtons Second Law says that the centripetal acceleration is accompanied by a force

F stands for any force that keeps an object following a circular path Force of friction (level and banked curves)Tension in a stringGravity

r

vmmaF C

2

FF

F

F

Net inward force to provide centripetal

acceleration

Due to contact and/or gravitational forces

Direction:

towards the center

CENTRIPETAL FORCE

SOME EXAMPLES

For a rock whirled on the end of a string, the

centripetal force is the force of tension in the string.

For an object sitting on a rotating turntable, the

centripetal force is friction.

For the motion of the Earth around the Sun, the

centripetal force is gravity.

Example

A small object of mass m is suspended from a

string of length L. The object revolves with constant

speed v in a horizontal circle of radius r. Find an

expression for v.

Example a:

Given:

String length: L

radius: r

Find:

1. v=?

1. Draw a free body diagram,

introduce coordinate frame and

consider vertical and horizontal

projections

r

vmT

2

sin r

v

g

TT

2cos

sin

2

cos

sinv

T

rgT

2(tan vrg

Newtons Law of Gravitation

1. Why are planets, moons and the

sun all nearly spherical?

2. Why do some of the earth

satellites circle the earth in 90

minutes while the moon takes 27

days for the trip?

3. Why dont satellites fall back the

earth?

For the motion of the Earth around the

Sun, the centripetal force is gravity.

Phenomenon of attraction between objects.

Modern physics describes gravitation using

Einsteins general theory of relativity, but the

much simpler Newtons Laws of Universal

gravitation provides an excellent

approximation in many cases.

RECALL : Newtons Third Law

If two objects interact, the force F12 exerted by

object 1 on object 2 is equal in magnitude but

opposite in direction to the force F21 exerted

by object 2 on object 1.

Equivalent to saying a single isolated force cannot

exist

Example: Newtons Third Law

Consider collision of

two spheres

F12 may be called the

action force and F21 the

reaction force

Actually, either force can

be the action or the

reaction force

The action and

reaction forces act on

different objects

Newtons Law of Universal Gravitation

Every particle in the Universe attracts every other

particle with a force that is directly proportional

to the product of the masses and inversely

proportional to the square of the distance

between them.

2

21

r

mmGF

G is the universal gravitational constant

G = 6.673 x 10-11 N m /kg

This is an example of an inverse square law

Gravitation Constant

Determined experimentally

Henry Cavendish

1798

The light beam and mirror serve to amplify

the motion

GRAVITATION

The reason for the very existence of most macroscopic objects in the universe

1. Heavenly Bodies Kept in Orbit

2. Heating the interiors of forming stars and planes to very high temperatures

3. Tides

4. Rising of Hot Air/Sinking of Cold Air (Convection) and other similar phenomenon

5. Keeping us in our toes!!!

JUMBO JUMBO

Find the gravitational force exerted by the Sun on the

Earth.

Me=6.0x1024kg;

Ms=2x1030 kg;

Res=1.5x108m;

G = 6.67 x 10-11 Nm2/kg2

Ans.3.56 x1028 N

Applications of Universal Gravitation :

Mass of the Earth

Use an example of an object close to the surface of the earth

r ~ RE

11 2

E

E

GM mm g

R

2

EE

gRM

G

Assumption:

RE + h RE

gmr

mGmF

e

eg 12

1

Gravitational Acceleration

mgr

mGmF

e

e

g 2

mgr

mGmF

e

e

g 2

re

h Assumption:

Re + h Re

2

e

e

r

Gmg

me

m

Applications of Universal Gravitation:

Acceleration Due to Gravity

g will vary with altitude

2 2

E EmM MF G m G mgr r

2

EMg Gr

Thus:

Weight is not an inherent property of an

object

mass is an inherent property

Weight depends upon location

Weight

The magnitude of the gravitational force acting

on an object of mass m near the Earths

surface is called the weight w of the object

w = m g is a special case of Newtons Second Law

g can also be found from the Law of Universal

Gravitation

The path of a satellite is an ellipse.

Circle is an ellipse with equal length of semi-major

and semi- minor axis.

r

vmma

r

mGmF

e

e

g

2

2

r

r

Gmv e

Applications of Universal Gravitation:

Motion of Satellite

Escape Speed

The escape speed is the speed needed for an

object to soar off into space and not return

For the earth, vesc is about 11.2 km/s

Note, v is independent of the mass of the object

2 Eesc

E

GMv

R

Every particle in the universe attracts every other particle with a

force that is directly proportional

to the product of the masses of the

particles and inversely proportional

to the square of the distance

between them.

Newtons Law of Universal Gravitation

Example

Question: Calculate gravitational attraction between two

students 1 meter apart. Assume the student 1 has a mass of

70 kg while the other one has a mass of 90 kg.

2111 2

22 2

70 906.67 10

1

mm N m kg kgF G

r kg m

74.2 10F N

Extremely small

compared to the

weight (F = mg).