Gravitation

The force with which the Earth pulls the objects towards it is called the gravitational force of
earth or Gravity .

Universal Law of Gravitation

Every object in the universe attracts other object by a force of attraction, called gravitation,
which is directly proportional to the product of masses of the objects and inversely proportional
to the square of distance between them. This is called Law of Gravitation or Universal Law of
Gravitation.

Let masses (M) and (m) of two objects are distance (d) apart. Let F be the attractional force
between two masses.

Importance of The Universal Law of Gravitation:

It binds us to the earth.

It is responsible for the motion of the moon around the earth.

It is responsible for the motion of planets around the Sun.

Gravitational force of moon causes tides in seas on earth.

Free Fall

The Falling off a body from a height towards the Earth under the gravitational force of earth
(with no other forces acting on it) is called free fall and such a body is called ‘freely falling body’.

 The acceleration produced in the freely falling bodies is the same for all the bodies and
it does not depend on the mass of the falling body .
 During free fall the heavier objects as well as the lighter objects accelerate at the same
rate.

Acceleration Due to Gravity (g):

When an object falls towards the earth there is a change in its acceleration due to the
gravitational force of the earth. So this acceleration is called acceleration due to gravity.

The acceleration due to gravity is denoted by g.

The unit of g is same as the unit of acceleration, i.e., ms−2

Mathematical Expression for g

From the second law of motion, force is the product of mass and acceleration.

F = ma

For free fall, acceleration is replaced by acceleration due to gravity.

Therefore, force becomes:

F = mg ….(i)

But from Universal Law of Gravitation,

To Calculate the Value of g:

Value of universal gravitational constant, G = 6.7 × 10–11 N m2/ kg2,

Mass of the earth, M = 6 × 1024 kg, and

Radius of the earth, R = 6.4 × 106 m

Putting all these values in equation (iii), we get:

Thus, the value of acceleration due to gravity of the earth, g = 9.8 m/s2.

Variation of acceleration due to gravity(g):

 The value of acceleration due to gravity ‘g’ ,is not constant at all the places on the
surface of the Earth. This is due to the fact that the Earth is not a perfect Sphere ,so the
value of its radius R is not same at all the places on its surface. In other words, due to
the flattening of the earth at the poles ,all the places on its surface or not at the same
distance from its centre and so the value of g varies with latitude. Since the radius of the
earth at the poles is minimum ,the value of g is maximum at the poles .again, the radius
of earth is maximum at the equator ,so the value of g is minimum at equator.
 The value of acceleration due to gravity ‘g’, is maximum on the surface of the earth, It
decreases ongoing above the surface of the earth or ongoing inside the surface of the
earth and it becomes zero at the centre of the earth.

Motion of Objects Under the Influence of Gravitational Force of the Earth (Equations of
motion for freely falling bodies ):

Let an object is falling towards earth with initial velocity u. Let its velocity, under the effect of
gravitational acceleration g, changes to v after covering the height h in time t.

Then the three equations of motion can be represented as:

Velocity (v) after t seconds, v = u + gt
Height covered in t seconds, h = ut + ½gt2
Relation between v and u excluding t, v2- u2 = 2gh

NOTE:

 When body is falling vertically downwards its velocity is increasing so the acceleration
due to gravity g is taken as positive.
  When a body is thrown vertically upwards its velocity is decreasing so the acceleration
due to gravity g is taken as negative.

 When a body is dropped freely from a height its initial velocity u is zero.

 When a body is thrown vertically upwards its final velocity V become zero.

 The time taken by a body to rise to the highest point is equal to the time takes to fall
from the same height.

Difference between Gravitation Constant (G) and Gravitational Acceleration (g)

Gravitational acceleration
S. No. Gravitation Constant (G)
(g)

1. Its value is 6.67×10-11Nm2/kg2. Its value is 9.8 m/s2.

2. It is a scalar quantity. It is a vactor quantity.

3. Its value remains constant Its value varies at various

always and everywhere. places.

4. Its unit is Nm2/kg2. Its unit is m/s2.

Mass & weight:

Mass (m)

The mass of a body is the quantity of matter contained in it.

Mass is a scalar quantity which has only magnitude but no direction.

Mass of a body always remains constant and does not change from place to place.

SI unit of mass is kilogram (kg).

Mass of a body can never be zero.

Weight (W)

The force with which an object is attracted towards the centre of the earth, is called the
weight of the object.
 Now, Force = m × a
But in case of earth, a = g
∴F=m×g
But the force of attraction of earth on an object is called its weight (W).
∴ W = mg

As weight always acts vertically downwards, therefore, weight has both magnitude and
direction and thus it is a vector quantity.

The weight of a body changes from place to place, depending on mass of object.

The SI unit of weight is Newton.

Weight of the object becomes zero if g is zero.

Weight of an Object on the Surface of Moon

Mass of an object is same on earth as well as on moon. But weight is different.

Weight of an object is given as,

Hence, weight of the object on the moon = (1/6) × its weight on the earth.

Thrust and Pressure

Thrust
 The force that acts in the perpendicular direction is called thrust.
 It is similar to force applied to an object
 It is a vector quantity.
Pressure
 The force that acts per unit area of the object is pressure.
 It is the thrust per unit area.
 Pressure is denoted by ‘P'
 P = thrust/ area = force/ area = F/A
 SI unit: N/m2 or Pa (Pascal)
Buoyancy
 Whenever an object is immersed in a liquid, the liquid exerts a buoyant force or
upthrust in the opposite direction of the gravitational force. This is also called the
Force of Buoyancy.
 It depends upon the density of the fluid.
 Therefore an object is able to float in water when the gravitational force is less than
the buoyant force.
 Similarly, an object sinks into the water when the gravitational force is larger than
the buoyant force.
 Figure 5 Buoyancy

Why does an object sink or float on water?

 An object can sink or float on water based on its density with respect to water. The
density is defined as mass per unit volume.
 Objects having a density less than water float in it. For Example, Cork flows in water
because its density is lower than that of water.
 Objects that have a density higher than water sink in it. For Example, Iron nails sink
in water because the density of iron is more than water's density.
 Thus, we can conclude that buoyancy depends upon:
 The density of the liquid
 The volume of the object (as the volume of object increases, its density decreases
and vice-versa)

Archimedes Principle
According to the Archimedes principle, whenever an object is immersed in a liquid (fully
or partially), the liquid exerts an upward force upon the object. The amount of that
force is equivalent to the weight of the liquid displaced by the object.
This means that if the weight of an object is greater than the amount of liquid it
displaces, the object will sink into the liquid. However, if the weight of an object is less
than the amount of water it displaces, the object will sink.
 Submarines have a tank called Buoyancy Tank. Whenever the submarine needs to be
taken inside water the tank is filled which thus increases the weight of the submarine.
 Similarly, when the submarine is to appear above water the tank is emptied and the
weight of the submarine becomes lighter and it rises above the water.
 Ships are heavier than water but their unique shape gives them a large volume. Their
volume is larger than their weight and hence the water displaced by a ship provides it
with the right upthrust so that it can float on water.

Applications of Archimedes Principle

 In evaluating relative density
 In designing ships and submarines
 In making lactometers and hydrometers

What is relative density?

When density can be expressed in comparison with water's density it is called Relative
Density. It has no unit because it is a ratio of two similar quantities.

Why is water chosen as a reference?

Water is present everywhere on earth so it becomes easier to evaluate the density of a
substance in relation to water.
How relative density can be used as a measure to determine if an object will sink or float
in water?
Relative Density of an object Float / Sink
Greater than 1 Sink in water
Less than 1 Float in water