Gravitation

1. GRAVITATION 01.
Earth must be attracting the any objects towards itself and this attractive force must be
directed towards the center of the earth.
Force (F) :
F= m x a ......(N) or (dyne)
1 N = 105 dyne
Force and Motion:
force is necessary to change the speed as well as direction of motion of an object.
Circular motion and Centripetal force:
Circular motion :
object moving along a circular path. This motion is called circular motion.

Centripetal force :
A force acts on any object moving along a circle and it is directed towards the centre of the
circle. This is called the Centripetal force.
F = m v2
r
m => mass of object, V => velocity of object, r => redius of circular path
`Centripetal’ means centre seeking, i.e. the object tries to go towards the centre of the circle
because of this force.
Newton’s universal law of gravitation:
According to this theory,
every object in the Universe attracts every other object with a definite force.




d

Statement :
This attraction force is directly proportional to the product of the masses of the two objects
and is inversely proportional to the square of the distance between them.
So,
F a (m1 x m2 )
d2
thus, F = G (m1 x m2 )
d2
G is the constant of proportionality and is called the Universal gravitational constant.
F => attraction force between two objects
m1, m2 => masses of the two objects
d => distance between two objects

, 02.
Kepler’s Laws:
Johannes Kepler stated three laws describing planetary motion. (planet revolce around the
sun)
A




Kepler’s first law :
The orbit of a planet is an ellipse with the Sun at one of the foci.
Kepler’s second law :
The line joining the planet and the Sun sweeps equal areas in equal intervals of time.
The straight lines AS and CS sweep equal area in equal interval of time i.e. area ASB and
CSD are equal.
Kepler’s third law :
The square of its period of revolution around the Sun is directly proportional to the cube of
the mean distance of a planet from the Sun.
Thus, if r is the average distance of the planet from the Sun and T is its period of revolution
then,
T2 < r3 i.e. T2 / r3 = constant = K
1)If the mass of one object is doubled, the force between the two objects also doubles.
F = G x m1m2 ......(1)
d2
F = G x { 2 x m1m2 } .........(2)
d2
F = 2 x G x m1m2
d2
F=2F ...............(from eq. (1) )

2)Also, if the distance is doubled, the force decreases by a factor of 4.
F = G x m1m2 ......(1)
d2
F = G x { m1m2 } .........(2) (distance is doubled)
2
(2d)
F = G x m1m2 = 1 x G x m1m2
4 x d2 4 d2

F = ( ) x F ...............(from eq. (1) )

i.e F= F/4

, 03.
3)If the two bodies are spherical, the direction of the force is always along the line joining
the centres of the two bodies and the distance between the centres is taken to be d.




d


4)In case when the bodies are not spherical or have irregular shape, then the direction of
force is along the line joining their centres of mass and d is taken to be the distance
between the two centres of mass.

5)In SI units, the value of G is equal to the gravitational force between two masses of 1 kg
kept 1 m apart.
In SI units its (G) value is
G = 6.673 x 10-11 N m2
kg2
6)The centre of mass of an object is the point inside or outside the object at which the total
mass of the object can be assumed to be concentrated.
7)The centre of mass of a spherical object having uniform density is at its geometrical
centre.
8)The centre of mass of any object having uniform density is at its centroid.
Uniform circular motion / Magnitude of centripetal force:
1) uniform circular motion :
When an object moves in a circular path with uniform speed, its motion is called uniform
circular motion.
2) Velocity of object in circular motion.
v = dispacement m
time taken s
3) If 'T' seconds required to complete one circular motion around the circular path of radius
r.
4) So, dispacement = circumference of circle
dispacement = 2 p r F
5) Velocity of object in circular motion becomes,

v = 2pr m
T s




distance = perimeter of the orbit = 2 x p x r