Transcript Tutorial Notes

GRAVITATION
Prepared by fRancis Chong
GRAVITATIONAL FIELD
It is a region in space that surrounds a mass, where its
influence can be observed.
In this region a force will be exerted on a mass that is
placed in the field.
GRAVITATIONAL FIELD
Field lines are drawn such that
- The tangent to the field represents the direction of g.
- The number of field lines per unit cross-sectional
area is proportional to the magnitude of g.
Earth
NEWTON’S LAW OF GRAVITATION
The force of attraction between two given point
masses is directly proportional to the product of their
masses and inversely proportional to the square of
their distance apart.
m1m2
F G 2
r
Sometimes written as
m1m2
F  G 2
r
To indicate that it is an attractive force.
NEWTON’S LAW OF GRAVITATION
Example
Find the gravitational force exerted by Earth on a point
mass at a distance h away from the Earth’s surface.
Solution
rE
Recall
m1m2
F G 2
r
h
ME m
F G
rE  h2
GRAVITATIONAL FIELD STRENGTH
It is defined as the gravitational force per unit mass
acting on a small mass placed at a point.
F
g
m
GRAVITATIONAL FIELD STRENGTH
Consider
r
M
From Newton’s Law of Gravitation,
m1m2
F G 2
r
Mm
G 2
GM
r
g 
 2
m
r
m
Also,
F
g
m
GRAVITATIONAL FIELD STRENGTH
Variation of g with distance from a point mass
GM
g 2
r
Taken from http://www.saburchill.com/physics/chapters/0007.html
GRAVITATIONAL FIELD STRENGTH
Variation of g with distance from the centre of a
uniform spherical mass of radius, R
r
M
m
Taken from http://www.saburchill.com/physics/chapters/0007.html
GRAVITATIONAL FIELD STRENGTH
Variation of g on a line joining the centres of two point
masses
If m1 > m2 then
Taken from http://www.saburchill.com/physics/chapters/0007.html
GRAVITATIONAL FIELD STRENGTH
This means that we can find a point between two
masses where their combined field strength is zero.
d
Moon
Earth
GM E
GM M

2
r2
d  r 
At this point, the net resultant gravitational force
exerted by both masses is zero. i.e. both the Earth and
Moon are exerting the same but opposite forces at p.
Taken from http://www.saburchill.com/physics/chapters/0007.html
GRAVITATIONAL POTENTIAL
The Gravitational Potential is defined as the work done
by an external agent in bringing a unit mass from
infinity to its present location.
GM

r
Unit of Φ = J kg-1
GRAVITATIONAL POTENTIAL

GM
r
The expression is negative because of the way we define
Gravitational potential.
- The zero of the potential is taken to be when the mass is
at infinity.
- Gravitational force is an attractive force.
- Therefore to bring a unit mass from infinity to its present
location, an external agent must be present to do
negative work.
GRAVITATIONAL POTENTIAL ENERGY
The Gravitational Potential energy is defined as the
work done by an external agent in bringing a mass
from infinity to its present location.
GMm
U
r
Unit of U = J
Tips for solving Orbital / Satellite problems
We always equate
GMm
mrw 
r2
2
or
2
mv
GMm

r
r2
ESCAPE VELOCITY
It is possible to find a velocity of a projected mass
such that it can escape the Earth’s gravitational field
and reach infinity.
At infinity, GPE is zero.
Also if the mass is just able to reach infinity, KE should
be zero.
Therefore total energy E = PE + KE = 0
By conservation of energy, total energy of the mass
should be the same throughout.
ESCAPE VELOCITY
Therefore, any object with total energy = 0 will be albe
to reach infinity.
E  PE  KE  0
 U T  0
GMm 1

 mv 2  0
R
2
2GM
v
 2gR
R
This is known as the escape velocity.
Practice 1
In the figure, ABC is a triangle in which CA = CB = r.
Equal masses m are situated at A and B.
(a)
In which direction relative to the x and y
directions does the resultant field at C act?
y
C
x
A
B
Practice
(b)
Find the gravitational potential at C.
y
C
x
A
B
Practice 2
At what height above the Earth’s surface will
g = 9.81 ms-1 rounded off to 2 d.p. change?