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• Gravitational fieldGravitational field• • Gravitational field strength (Gravitational field strength (gg))

6.2 Gravitational field and 6.2 Gravitational field and gravitational field strengthgravitational field strength

• • Relationship between Relationship between GG and and gg• • Variation of Variation of gg with a distance with a distance

from the centre of earthfrom the centre of earth• • Variation of Variation of g g with latitudewith latitude


Gravitational field

6.2 Gravitational field and gravitational field strength (SB p. 203)

A gravitational field is a force field. This field exists around a mass. Another mass introduced into the field will experience a gravitational force.

Similar to magnetic field

earth’s gravitational field represented by

lines of forces


Gravitational field


Gravitational field- only the force of attraction- directed towards centre of earth- stronger as closer to earth’s surface

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More to Know 3More to Know 3

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Gravitational field strength (g)


Unit – N kg-1

The gravitational field strength (g) at a point in a gravitational field is defined as the gravitational force of attraction per unit mass at that point.

mFg

g

Mass

force nalGravitatio )(strength field nalGravitatio

vector


Gravitational field strength (g)


By Newton’s 2nd Law

which is equal to g (acceleration due to gravity)

mFonAccelerati

Unit of gravitational field strength = Unit for acceleration

Gravitational field strength on earth’s surface= 9.81 N kg-1 (produces acceleration of 9.81 m s-2)


Relationship between G and g


2

2

By

E

E

E

E

R

MGg

mgR

mMGF

1. Mass of earth (ME) = kg 105.99 242

GgRE

2. Average density of earth ()

=3-3 m kg 10505 .

VME

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Example 2Example 2

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Example 3Example 3

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Example 4Example 4


Variation of g with a distance from the centre of earth


1. For particles on or above the earth’s surface

(a) At r (r RE)

2

2

2

r

GMmF'gr

GMmF

r

mMGF

E

E

E




1. For particles on or above the earth’s surface

(b) r = RE

1-2o kg N 89.

R

GMg

E

E

22

2

o

2

2

o

1 rr

Rg'g

r

Rg'g

E

E

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Example 5Example 5




2. For particles below the earth’s surface

Gravitational force acted on particle is only due to part of earth inside it

rRrg'g

r'GM'gr'mMG'mg

MRr'M,

R

r

M'M

Eo

EE

EE

2

2

3

3

3

iton acted force nalGravitatio particle ofWeight

3434

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2. For particles below the earth’s surface

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Variation of g with latitude


Two main reasons why g varies with latitude

• Shape of earth- earth is ellipsoid, R1 R2

• Self-rotation of earth- all objects on earth’s surface (except at the poles) need centripetal forces




1. Gravitational field strength at the poles gp

2E

Ep

R

MGgg




2. Gravitational field strength at the equator ge

2

22

22

Epe

peEE

E

eEE

E

Rgg

ggRR

MG

mgmRR

mMGF

1-2

2

kg N 10373

.

Rgg Eep

ge < gp




3. Gravitational field strength at latitude (g)

Centripetal force (towards Q)= m(RE cos)2

which is provided by components of gravitational force (towards O) and other force from object

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3. Gravitational field strength at latitude (g)

22

o

2o

cos coscoscos

E

E

RggRmmgmg

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If = 45o

g = 9.79 N kg-1

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Radial and uniform gravitational fields

On the earth or other planets, the gravitational field lines are radially inward (radial field). The parallel field lines indicate that it is a uniform field.

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Uniform gravitational field

If a body is located in a uniform gravitational field, the gravitational force acted on the body is the same for all positions.

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Uniform gravitational field near the earth's surface

Since the radius of the earth is very great, the gravitational field near the earth's surface is assumed as uniform.

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g for an isolated sphere

Other than the earth, the gravitational field strength (g) of a point at a distance r from the centre of an isolated sphere of mass M is also equal to:

2rMGg

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Q:Q: The mass of the earth is approximately 80 times that of the moon and the earth’s radius is 3.7 times that of the moon. If the acceleration of free fall on the earth is 9.81 m s−2, estimate the gravitational field strength (g) on the moon’s surface.

Solution



Solution:Solution:

2-

2

2

2

2

221

s m 7180

73819

7380 :

(2)(1)

moon. on the fall) free ofion (acceleratstrength field nalgravitatio' where

surface, smoon' on the mass same For the73

80

surface, searth' on the mass aFor

By

...

'g,.'g

gg

rmMG'mg

mr.

MmGmg

mr

mmGF

If M = mass of moon, then mass of earth = 80M ;if r = radius of moon, then radius of earth = 3.7r.


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Q:Q: A space capsule travels from the earth towards the moon. Calculate its distance from the earth where it is under zero gravity. (Consider only the gravity due to the earth and the moon.)

(Mass of earth ME = 5.98 × 1024 kg; mass

of moon MM = 7.35 × 1022 kg; distance

between centre of earth and centre of moon = 3.84 × 108 m.) Solution



Solution:Solution:

m 10463

02910843

1035710985

10843

10843

8

8

22

242

8

282

21

.x

.x.

x..

MM

x.x

x.

mMG

x

mMG

gg

M

E

ME

Let x be the distance of space capsule from the earth when it is under zero gravity. This occurs if


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Q:Q: A certain star of mass M and radius r rotates so rapidly that materials at its equator only just remain on its surface. Given that the gravitational constant is G, find the period of self-rotation of the star in terms of G, M and r.

Solution



Solution:Solution:

2rmMGF

For a mass m on the equator, the gravitational force:

As the star rotates, the mass moves in a circle of radius r. The centripetal force is provided by the gravitational force.


GMrT

rGM

mrrmMG

3

3

22

22)( Period

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Q:Q: The radius of the earth (RE) is 6.38 × 106

m and the acceleration due to gravity at the earth’s surface is 9.81 m s−2. Find the acceleration of free fall at a height of 6.38 × 104 m from the earth’s surface.

Solution



Solution:Solution:

The distance from the earth’s centre (r) = 6.38 ×104 + 6.38 ×106 = 6.44 ×106 m = 1.01RE

Acceleration of free fall at the height ( g’)


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s m 6290111819

0112-

22

2

o2

2

o ..

.R.

Rg

r

Rg

E

EE


1. In the case of a shell and a particle outside the shell, the shell attracts the particle as if all the mass of the shell were concentrated at its centre.

2. If the particle is inside the shell, the net force acted on the particle is zero.


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g at earth’s centre

At the earth’s centre, r = 0.

Hence, g = 0.

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Centripetal force

In the calculation, the gravitational force is assumed to be the same everywhere on the earth. Therefore, the centripetal force is affected by latitude only.

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If a plumb-line is hung at P, it will not be exactly normal to the earth’s surface but will be along the direction PB.

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g on the earth

Generally, the gravitational field strength varies from place to place on the earth's surface. It increases from the equator to the poles.

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