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Topic 6: Fields and Forces
6.1 Gravitational force and field
Newton’s universal law of gravitation
• Newton proposed that a force of attraction
exists between any two masses.
• This force law applies to point masses not
extended masses
• However the interaction between two
spherical masses is the same as if the
masses were concentrated at the centres
of the spheres.
Newton´s Law of Universal Gravitation
• Newton proposed that
“every particle of matter in the universe
attracts every other particle with a force
which is directly proportional to the
product of their masses, and inversely
proportional to the square of their distance
apart”
• i.e.
F  m1m2
&
1
F 2
r
• Where m1 and m2 are the masses of the 2
objects and r is the distance between them
• This can be written as
• F = G m1m2
r2
• Where G is Newton´s constant of
Universal Gravitation
• It has a value of 6.67 x 10-11 Nm2kg-2
Define gravitational field strength.
• A mass M creates a gravitational field in
space around it.
• If a mass m is placed at some point in
space around the mass m it will
experience the existance of the field in the
form of a gravitational force
• We define the gravitational field strength at
a point in a gravitational field as the force
acting on a 1kg mass placed at that point.
• The force experienced by a mass m
placed a distance r from a mass M is
• F = G Mm
r2
• And so the gravitational field strength of
the mass M is given by dividing both sides
by m
• g=GM
r2
• The units of gravitational field strength are
N kg-1
• The gravitational field strength is a vector
quantity whose direction is given by the
direction of the force a mass would
experience if placed at the point of interest
Variation of g with distance from
a point mass, M
• Gravitational field strength, g = GM/r²
Variation of g with distance from the centre
of a uniform spherical mass of radius, R
Variation of g on a line joining
the centres of two point masses
• s
• In order to calculate the overall
gravitational field strength at any point, we
must use vector addition.
• The overall gravitational field strength at
any point b/w the Earth and the Moon
must be a result of both pulls.
• g = Fresultant/m
• There will be a single point somewhere
b/w the Earth and the Moon where the
total gravitational field due to these two
masses is zero.
Planet of mass M
Radius R
Relying on the fact that the
sphere behaves as a point
mass situated at its centre,
from Newton’s Law of
gravitation,
For an object of mass m on
the surface of the planet, the
gravitational force is
F = G Mm
R2
Field Strength at the
Surface of a Planet
• Since gravitational field strength is defined
as the gravitational force acting on 1kg of
mass,
• The field strength on the surface of the
planet is
• g= GM
R2
• If the planet is the Earth then we have the
gravitational field strength on the surface of
Earth as
• g = G Me
R e2
• where Me and Re are the mass and radius
of the earth respectively.
• Me = 6.0 x 1024 kg
• Re = 6.4 x 106 m
What is the relation between the Acceleration due
to Gravity and the Universal Constant of
Gravitation?
• From newton’s 2nd law, F = mg
• We also know F = G Mem
•
R e2
• hence we have that the acceleration of
free fall at the surface of the Earth, g
• g = G Me
R e2