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Mechanics
Topic 9.2 Gravitational field,
potential and energy
Gravitational Force and Field
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 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”
This can be written as
F = G m1m2
r2
Where G is Newton´s constant of
Universal Gravitation
It has a value of 6.667 x 10-11 Nm2kg-2
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 as the ratio of the force the
mass m would experience to the mass,
m
That is the gravitational field strength at
a point, it is the force exerted per unit
mass on a particle of small 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
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
Field Strength at the
Surface of a Planet
If we replace the particle M with a
sphere of mass M and radius R then
relying on the fact that the sphere
behaves as a point mass situated at its
centre the field strength at the surface
of the sphere will be given by
g= GM
R2
If the sphere is the Earth then we have
g = G Me
Re2
But the field strength is equal to the
acceleration that is produced on the mass,
hence we have that the acceleration of free
fall at the surface of the Earth, g
g = G Me
Re2
Gravitational Energy
and Potential
We know that the graviational potential
energy increases as a mass is raised
above the Earth
The work done in moving a mass
between two points is positive when
moving away from the Earth
By definition the gravitational potential
energy is taken as being zero at infinity
The gravitational potential at any point in the
Earth´s field is given by the formula
V = - G Me
r
Where r is the distance from the centre of the
Earth (providing r >R)
The negative sign allows for the fact that all
the potentials are negative as they have to
increase to zero
Definition
The potential is therefore a measure of the
amount of work that has to be done to move
particles between points in a gravitational
field and its units are J kg –1
The work done is independent of the path
taken between the two points in the field, as
it is the difference between the initial and
final potentials that give the value
Escape Speed
The escape speed is the speed required
for a projectile to leave the Earth´s
gravitational attraction.
i.e. To get to infinity!
If the potential at the Earth´s surface is
V = - G Me
Re
Then the Ep change to get to infinity is
G Me x m
Re
Where m is the mass of the projectile
For this amount of energy to be gained the
projectile must have had an equal amount of
Ek
Therefore ½mv2 = G Me x m
v = ( 2GMe
Re )
Re
But using the fact that g = G Me
Then v = (2gRe )
Re2