Unit – III A |

## Concept of Inertial Mass and Gravitational Mass

### Inertial Mass

#### Concept:

- The inertial mass of a body is related to its inertia in linear motion.
- Let a body of mass ‘m’ moves with an acceleration ‘a’ along a straight line under the action of force ‘F’. Then By Newton’s second law of motion,

- Let a = 1 then m = F. Thus the inertial mass of a body is equal to the magnitude of external force required to produce unit acceleration in the body.

#### Characteristics of Inertial Mass:

- Inertial mass is a measure of the inertia of a body and is proportional to the quantity of matter contained in the body.
- The inertial mass of a body is independent of size, shape and state of a body.
- It is not dependent on the temperature of the body.
- It is not affected by presence or absence of other bodies near it.
- It obeys the simple algebraic law of addition when combined and the simple algebraic law of subtraction when separated or removed.
- In a chemical reaction, the inertial mass of a substance is conserved.
- The inertial mass of a body increases with its speed. The new inertial mass is given by

Where ‘m_{O}’ is rest mass of a body, ‘c’ is the speed of light and ‘v’ is the speed of the body.

- Inertial mass is affected only if the speed of the body is comparable with that of light.

### Gravitational Mass:

#### Concept:

- The gravitational mass of a body is related to gravitational pull on the body.
- If body of mass ‘m’ is at rest on the surface of the Earth of radius ‘R’ and mass ‘M’, then gravitational force on the body is given by

Where ‘E’ is gravitational intensity.

- Let E = 1, then m = F. Thus, the gravitational mass of a body is defined as the magnitude of a gravitational pull experienced by the body in a gravitational field of unit intensity.

- The characteristics of gravitational mass are the same as that of inertial mass.

### Comparison of Inertial Mass and Gravitational Mass:

- Both are scalar quantities and have same units of measurement.
- Gravitational mass of a body is affected by presence or absence of other bodies near it. Whereas inertial mass is not affected by presence or absence of other bodies near it.
- The inertial mass of a body is calculated by finding acceleration in the body and the force causing the acceleration without considering gravity. Whereas gravitational mass of a body is calculated by finding gravitational pull on the body without considering the acceleration in the body and the force causing the acceleration.
- The gravitational mass is measured by spring balance and inertial mass is measured by an inertial balance.

### Equivalence of Inertial Mass and Gravitational Mass:

- Let us consider two bodies say A and A’ of gravitational masses M
_{g}and M_{g}’ respectively. Let They are kept in the gravitational field of Erath of gravitational mass ‘M’ at equal distance ‘R’ from the Earth. Then the force on body A is given by

- Let M
_{i}and M_{i}’ respectively be their inertial masses of the two bodies. Now let the two bodies allowed to fall downward in the vacuum from the same height under gravity, where the acceleration due to gravity is ‘g’.

- Thus the gravitational mass of a body is directly proportional to its inertial mass. i.e. gravitational mass and inertial mass are equivalent.
- This equivalency concept is used by Einstein to develop his Theory of Relativity.