Download Gravitation

1
Gravitation-only an attractive force!
Why doesn’t the atmosphere or air just escape into outer space, leaving us breathless?
Why are we bound to the solar system? Why is the moon bound to the earth at such a long distance?
Newton’s theory of universal gravitation answers these questions.
You are aware that a positively charged proton attracts a negatively charged electron because of the
interaction between their electric charges. You are also aware that an electron repels another electron
because of the interaction between their electric charges. A magnetic north pole attracts a south magnetic
pole, but repels a north pole. These are electromagnetic interactions: they can be attractive or repulsive.
Now the earth attracts the moon. This is the reason the moon moves in a circular orbit around the earth.
The earth also attracts all other objects - this is the reason objects fall to the ground. A piece of wood,
which is neither an electrically charged particle nor a magnet, falls to the ground. Thus, the pull of the
earth on the piece of wood cannot be electromagnetic.
The force of attraction between a piece of wood and the earth is a new kind of force: gravitational force.
This attractive force arises due to the interaction between the neutral mass of one object and the neutral
mass of the other. The neutral mass of an object is called its gravitational charge. Electrical charge is
measured in coulombs, but gravitational charge is measured in kg. However, the simple word for this
gravitational charge is mass (kg)
Newton’s law of universal gravitation: Two objects attract each other with a force which
is proportional to the product of their masses, and inversely proportional to their
separation.
F
m1m2
F 

d2
d
m1
m2
m1m2
F G 2
d
Minus indicates the vector force, F, is attractive. D is the centre to centre separation
between the two objects, and m represents the mass in kg.
As long as you are aware of the attractive nature of the gravitational force, we can do
solve most of the problems using the magnitude of the force:
mm
F  G 1 2 2 (Magnitude of force)
d
G is called the universal gravitational constant. The unit of G can be found from:
 d2 
2
2
G  F
 = Nm kg
 m1m2 
The magnitude of G is very small and approximately equal to 6.62x10 -11.
Thus, G = 6.62x10 -11 Nm2 kg 2 .
2
Gravitational force is a weak force, and only becomes significant when at least one of the masses is very
large – like a planet.
Example: Calculate the force between two objects, each 1 kg, and 1 metre apart.
F = 6.62 x 10 -11 * (1 x1) / 12 = 6.62x10 -11 N. This is very, very small.
Example: Calculate the force between two objects, one 1 X 10 12 and the other 1 kg, separated 1m.
F = 6.62 x 10 -11 * (1012 x1) / 12 = 66.0 N.
Gravitational field
Why does one object exert a gravitational attractive force on another object?
(a) Why does m1 attract m2? (b) Why does m2 attract m1? These are different questions
and require different answers.
Let us consider an analogy first: consider a girl wearing some expensive perfume. This
material gives the space around her a very special, fragrant, quality. When another
person, say, a boy, enters this fragrant field, the fragrance, or more exactly the fragrant
field, interacts with the boy and applies an attractive force on him, which is towards the
girl. There are four aspects to this example: The girl, the attractive fragrant field, the
boy, and the interaction between the fragrant field and the boy that results in an
attractive force on the boy.
Similarly, m1 creates an attractive gravitational (not fragrant) field around itself. When
another object, m2, enters this field, the ‘gravitational field’ interacts with m2, and
exerts an attractive force on it.
But how does m2 attract m1?
m2 creates an attractive gravitational field around itself. When another object, m1,
enters this field, the field interacts with m1, and exerts an attractive force on it.
(Note carefully: we do not have one gravitational field interacting with another gravitational field to producing forces on masses.)
Three questions arise about the gravitational field.
(a) What does it look like?
(b) How can we measure the strength of this field?
(c) How can we calculate the strength of this field?
(a) Suppose we want to observe the gravitational field of the earth, an object which is quite massive,
let a small object, e.g. a brick fall to the earth. The path of the brick as it moves towards the earth
represents Earth’s gravitational field. To get a three dimensional picture of the field, you will have
to release the brick from different points around the earth. Note that the gravitational field has a
direction – the direction in which the brick falls. Therefore the gravitational field is a vector field.
(b) The strength of the field at a particular point is defined as the force on 1 kg of mass. Or, put a
mass m kg at a point, measure the force on it, and then divide the measured force my m.
(c) How can we calculate it? Since gravitational field strength is defined as the force on I kg, we can
use the force formula but replace m2 by 1
kg. Thus G.F.S. =  G
mE *1
m
  G E2
2
d
d
3
We can write the Newton’s gravitational force formula in two ways:
mm
m 

F   G 1 2 2 , F   G 21  * m2 = GFS of mass 1 * mass2. (= force on m2)
d
d 

m 

GFS of mass 1   G 21 
d 

Is there any evidence that Newton’s theory of universal gravitation is true?
The word ‘true’ is not appropriate for scientific theories:
We understand observations of the universe around us in terms of our theories. If there are carefully
observed events that cannot be explained by our theory, then our theory can be said to have been falsified.
If on the other hand, our theory is able to explain all carefully observed events, then we can say that our
theory is quite sound, but not necessarily true as there may be events in the future (not yet observed) that
our theory may not be able to explain. Thus all theories, no matter how sound, are provisional.
Is Newton’s theory of gravitation a sound theory?
Newton’s theory gave a spectacular explanation of Kepler’s laws, and established itself as a sound theory.
Kepler’s laws: In our solar system
(1) Each planet orbits the sun in an elliptical path, the sun being at one of the foci of the elliptical
orbit.
(2) Each planet orbits the sun in such a way that the line joining the planet and the sun sweeps out
equal areas in equal times.
(3) The square of the period of a planet divided by the cube of its average distance from the sun is the
same for all planets, or.
T2 
r 3 , or : T 2  Kr 3
( K  constant) .
Newton’s Explanation of Kepler’s third law:
v =Velocity
Planet
R
SUN
Suppose a planet is orbiting the sun in a circular (a special case of ellipse) orbit.
Let the radius of orbit = R
Let the velocity of the planet be v
Let the mass of the sun = M
Let the mass pf the planet be m
The planet requires a centripetal force (towards the sun) =
mv 2
R
This force is provided by the gravitational pull of the sun = F = G
Therefore,
Mm
R2
4
2
Mm
M
mv
= G 2  v2  G
R
R
R
Now period =
T
(1)
2 R
2 R
4 2 R 2
(2)
v 
 v2 
v
T
T2
Now comparing equations 1 and 2 =>
 4 2  3
M 4 2 R 2
2
G

T  
R T 2 

2
R
T
 GM 
R3
Thus Newton’s law explains Kepler’s observations.
Newton’s theory of gravitation is a very powerful theory, but it is not a true theory.
Einstein has given a more powerful explanation of gravity in his General Theory of
relativity, but this is also not a true theory – only a more powerful theory for it explains
everything that Newton’s theory explains and much more.
***




Still to come:
Verification of Kepler’s law (Data analysis)
Gravitational potential energy
Applications of Newton’s theory to the solar system
Geostationary orbits. Weightlessness