Physics

Gravitation

Session

Session Opener

“Planets revolve in elliptical orbits with sun at its focus”. Have you ever wondered what forces are responsible?

Session Objectives

Session Objective

Newton's law of Gravity

Weight & Gravitational Force

Kepler's Law’s

Gravitational Field

Gravitational Potential

Relation between field and potential

Gravitational Potential Energy

Any particle of mass m1 attracts a particle of mass m2 with a force given by:

221

r

mmGF

Where,

r is the distance between them

G is Universal Gravitational Constant

G=6.67x10-11 N.m2/kg2

m1 m2F

r

Newton’s Law of Gravitation

Newton’s Law of Gravitation

Important fact about this formula

• Applicable only for point masses

• Does not depend on the medium between the masses

• Not valid for nuclear distances

• These forces are always added vectorially

Weight & Gravitational force Earth attracts every body towards itself by virtue of which the body experience weight. The true weight of nay body is only at the earths surface at every other place it has apperant weight.

Weight = mass x acceleration due to gravity.

W = mg

Kepler’s Laws

Kepler’s First Law:

Kepler’s Second Law:

All planet move in elliptical orbits with the sun at one focus.

A line that connects a planet to the sun sweeps out equal areas in equal time, i.e. the areal velocity of the planet is always constant

Sun

Kepler’s Laws

Kepler’s Third Law:

dAcons tant

dt

2 3T a

The square of the period of revolution of any planet is proportional to the cube of the semi-major axis of the orbit.

Sun

A

B

A’

B’

m

FE [Gravitational Field]

Gravitational Field Intensity

It is defined as force experienced per unit mass acting on a test mass supposed to be placed at that point.

2 2

GMm 1 GME

mr r Source

point

Fieldpoint

m (Test mass)

M

r

[Gravitational Potential]

Gravitational Potential

It is defined as negative of the work done per unit mass in shifting a rest mass from some reference point to the given point.

WV

m

UV

m

GMV(r)

r

Relation between Gravitational Potential

F mE

dW F.dr

mE.dr.GGGGGGGGGGGGG G

dU dW mE.dr. GGGGGGGGGGGGG G

dUdV E.dr.

m

GGGGGGGGGGGGG G

M

U

F(r) m

r

r

GMm

[For a point mass]

Gravitational Potential Energy

GMmU(r)

r

Gravitational potential energy at a point is defined as the amount of work done by an external agent in bringing any body of mass (m) from infinity to that point.

Expressions of potential for different bodies

Gravitational potential V due to a spherical shell of mass M and radius R at a point distant r from the centre.

(a) When r > R

GM

Vr

(b) When r = R

GM

VR

(c) When r < R

GM

VR

(d) When r = 0

GM

VR

V

R r

Expressions of potential for different bodies

Gravitational potential V due to a solid sphere of radius R and mass M at a point distant r from the centre.

(a) When r > R

GM

Vr

(b) When r = R

GM

VR

(c) When r < R (d) When r = 0

2 2

3

3R rV GM

2R

3 GMV

2 R

Expressions of gravitational field for different bodies

Gravitational field E due to a spherical shell of mass M and radius R at a point distant r from the centre.

(a) When r > R

2

GME

r

(b) When r = R

2

GME

R

(c) When r < R

E = 0

(d) When r = 0

E = 0

rR

E

Expressions of gravitational field for different bodies

Gravitational field E due to a solid sphere of radius R and mass M at a point distant r from the centre.

(a) When r > R (b) When r = R

2

GME

r

2

GME

R

(c) When r < R (d) When r = 0

E = 0

3

GMrE

RE

R r

Class Test

Class Exercise - 1

In order to find time, an astronaut orbiting in an earth satellite can use

(a) pendulum clock

(b) spring -on trolled clock

(c) any one of above

(d) Neither of the two

Solution

As the acceleration for a satellite continuously changes so it will give wrong time. Where this is not in case of spring-controlled clock.

Hence answer is (b).

Class Exercise - 2

Which of the following graphs represent the motion of a planet moving about the sun? T is the period of revolution and r is the average distance (from centre to centre) between the sun and the planet.

T 2

r3

(a)

T2

r3

(b)

r

T 2

(c) T 2

r3

(d)

Solution

By statement of Kepler’s law

Hence answer is (a).

Class Exercise - 3

A planet of mass M moves around the sun along an ellipse so that its minimum distance from the sun is r and maximum is R. Using Kepler’s law, find its period of revolution around the sun.

Solution

According to Kepler’s law

32 3r R

T K Kx2

R rwhere x and K Cons tant

2

2

s2

GMMMvAlso

x x 2 s sGM GM

v ; vx x

s

2 x 2 xT

v GM

x

3 / 2

s

2 r R

2GM

Class Exercise - 6

If the radius of the earth were to shrink by 1%, its mass remaining the same, the acceleration due to gravity on the earth’s surface would

(a) decrease (b) remain unchanged

(c) increase (d) Cannot say

Solution

2

GMg

R

but as R is decreased so g would increase.

Hence answer is (c).

Class Exercise - 7

Two planets of radii r1 and r2 are made from the same material. The ratio of the

acceleration of gravity at the surfaces

of two planets is

1

2

g

g

1 2

2 1

2 21 2

2 1

r r(a) (b)

r r

r r(c) (d)

r r

Solution

31

1 21

4G r

3g2r

14

G r3

1 1

2 2

g rso,

g r

Hence answer is (a).

Thank you