FREE FALL

We have learnt that the earth attracts
objects towards it. This is due to the
gravitational force. Whenever objects fall                        

                                                             

towards the earth under this force alone, we
say that the objects are in free fall. Is there
any change in the velocity of falling objects?
While falling, there is no change in the
direction of motion of the objects. But due to
the earth’s attraction, there will be a change
in the magnitude of the velocity. Any change
in velocity involves acceleration. Whenever an
object falls towards the earth, an acceleration
is involved. This acceleration is due to the
earth’s gravitational force. Therefore, this
acceleration is called the acceleration due to
the gravitational force of the earth (or
acceleration due to gravity). It is denoted by
g. The unit of g is the same as that of
acceleration, that is, m s–2.
We know from the second law of motion
that force is the product of mass and
acceleration. Let the mass of the stone in
activity be m. We already know that there
is acceleration involved in falling objects due
to the gravitational force and is denoted by g.
Therefore the magnitude of the gravitational
force F will be equal to the product of mass
and acceleration due to the gravitational
force, that is,
F = m g
m g
d
or G 2
M
g =
d

where M is the mass of the earth, and d is
the distance between the object and the earth.
Let an object be on or near the surface of
the earth. The distance d  will be
equal to R, the radius of the earth. Thus, for
objects on or near the surface of the earth,
G 2
M × m
mg =
R

G 2
M
g =
R

The earth is not a perfect sphere. As the
radius of the earth increases from the poles
to the equator, the value of g becomes greater
at the poles than at the equator.

How did Newton guess the inverse square law?

There has always been a great interest
in the motion of planets. By the 16th
century, a lot of data on the motion of

                                                                 

planets had been collected by many                                
astronomers. Based on these data
Johannes Kepler derived three laws,
which govern the motion of planets.
These are called Kepler’s laws. These are:
1. The orbit of a planet is an ellipse with
the Sun at one of the foci, as shown in
the figure given below. In this figure O
is the position of the Sun.
2. The line joining the planet and the Sun
sweep equal areas in equal intervals
of time. Thus, if the time of travel from
A to B is the same as that from C to D,
then the areas OAB and OCD are
equal.
3. The cube of the mean distance of a
planet from the Sun is proportional to
the square of its orbital period T. Or,
r3/T2 = constant.
It is important to note that Kepler
could not give a theory to explain
the motion of planets. It was Newton
who showed that the cause of the
planetary motion is the gravitational
force that the Sun exerts on them. Newton
used the third law
of Kepler to
calculate the
gravitational force
of attraction. The
gravitational force
of the earth is
weakened by distance. A simple argument
goes like this. We can assume that the
planetary orbits are circular. Suppose the                
orbital velocity is v and the radius of the
orbit is r. Then the force acting on an
orbiting planet is given by F v2/r.
If T denotes the period, then v = 2πr/T,
so that v2 r 2/T2.
We can rewrite this as v2 (1/r) ×
( r3/T2). Since r3/T2 is constant by Kepler’s
third law, we have v2 1/r. Combining
this with F v2/ r, we get, F 1/ r2.

UNIVERSAL LAW OF GRAVITATION

Every object in the universe attracts every
other object with a force which is proportional             
to the product of their masses and inversely
proportional to the square of the distance
between them. The force is along the line
joining the centres of two objects.

GRAVITATION

We know that the moon goes around the
earth. An object when thrown upwards,
reaches a certain height and then falls
downwards. It is said that when Newton was              
                                                                                 
sitting under a tree, an apple fell on him. The
fall of the apple made Newton start thinking.
He thought that: if the earth can attract an
apple, can it not attract the moon? Is the force
the same in both cases? He conjectured that
the same type of force is responsible in both
the cases. He argued that at each point of its
orbit, the moon falls towards the earth,
instead of going off in a straight line. So, it
must be attracted by the earth. But we do
not really see the moon falling towards the
earth.
The motion of the moon around the earth
is due to the centripetal force. The centripetal
force is provided by the force of attraction of
the earth. If there were no such force, the
moon would pursue a uniform straight line

motion.
It is seen that a falling apple is attracted
towards the earth. Does the apple attract the
earth? If so, we do not see the earth moving
towards an apple. Why?
According to the third law of motion, the
apple does attract the earth. But according
to the second law of motion, for a given force,
acceleration is inversely proportional to the
mass of an object. The mass of an
apple is negligibly small compared to that of
the earth. So, we do not see the earth moving
towards the apple. Extend the same argument
for why the earth does not move towards the
moon.
In our solar system, all the planets go
around the Sun. By arguing the same way,
we can say that there exists a force between
the Sun and the planets. From the above facts
Newton concluded that not only does the
earth attract an apple and the moon, but all
objects in the universe attract each other. This
force of attraction between objects is called the
Gravitational force.

WORLD IN PHYSICS

“Time and space are finite in extent, but they don't have any boundary or edge. They would be like the surface of the earth, but with two more dimensions.” 
                                                                                                                Stephen Hawking.