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© Manhattan Press (H.K.) Ltd. 1
• 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
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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
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Gravitational field
6.2 Gravitational field and gravitational field strength (SB p. 203)
Gravitational field- only the force of attraction- directed towards centre of earth- stronger as closer to earth’s surface
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Gravitational field strength (g)
6.2 Gravitational field and gravitational field strength (SB p. 203)
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
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Gravitational field strength (g)
6.2 Gravitational field and gravitational field strength (SB p. 204)
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)
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Relationship between G and g
6.2 Gravitational field and gravitational field strength (SB p. 204)
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
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Variation of g with a distance from the centre of earth
6.2 Gravitational field and gravitational field strength (SB p. 208)
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
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Variation of g with a distance from the centre of earth
6.2 Gravitational field and gravitational field strength (SB p. 208)
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
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Variation of g with a distance from the centre of earth
6.2 Gravitational field and gravitational field strength (SB p. 209)
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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Variation of g with a distance from the centre of earth
6.2 Gravitational field and gravitational field strength (SB p. 210)
2. For particles below the earth’s surface
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Variation of g with latitude
6.2 Gravitational field and gravitational field strength (SB p. 210)
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
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Variation of g with latitude
6.2 Gravitational field and gravitational field strength (SB p. 211)
1. Gravitational field strength at the poles gp
2E
Ep
R
MGgg
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Variation of g with latitude
6.2 Gravitational field and gravitational field strength (SB p. 211)
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
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Variation of g with latitude
6.2 Gravitational field and gravitational field strength (SB p. 212)
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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Variation of g with latitude
6.2 Gravitational field and gravitational field strength (SB p. 212)
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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End
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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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6.2 Gravitational field and gravitational field strength (SB p. 203)
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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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6.2 Gravitational field and gravitational field strength (SB p. 203)
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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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6.2 Gravitational field and gravitational field strength (SB p. 204)
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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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6.2 Gravitational field and gravitational field strength (SB p. 204)
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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
6.2 Gravitational field and gravitational field strength (SB p. 205)
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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.
6.2 Gravitational field and gravitational field strength (SB p. 205)
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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
6.2 Gravitational field and gravitational field strength (SB p. 205)
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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
6.2 Gravitational field and gravitational field strength (SB p. 206)
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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
6.2 Gravitational field and gravitational field strength (SB p. 206)
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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.
6.2 Gravitational field and gravitational field strength (SB p. 206)
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
6.2 Gravitational field and gravitational field strength (SB p. 209)
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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’)
6.2 Gravitational field and gravitational field strength (SB p. 209)
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s m 6290111819
0112-
22
2
o2
2
o ..
.R.
Rg
r
Rg
E
EE
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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.
6.2 Gravitational field and gravitational field strength (SB p. 209)
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g at earth’s centre
At the earth’s centre, r = 0.
Hence, g = 0.
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6.2 Gravitational field and gravitational field strength (SB p. 210)
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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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6.2 Gravitational field and gravitational field strength (SB p. 212)
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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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6.2 Gravitational field and gravitational field strength (SB p. 212)
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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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6.2 Gravitational field and gravitational field strength (SB p. 213)
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