Mass And Weight

1.0Definition of Mass

The mass of a body refers to the amount of matter it contains, making it a fundamental property of matter. This quantity remains constant regardless of temperature, pressure, or the body's location in space. The SI unit for mass is the kilogram (kg).

2.0Types of Mass(Inertial And Gravitational Mass)

1. Inertial Mass

• It measures its inertia is called Inertial Mass.
• It is equal to the ratio of the external force applied on the body to the acceleration produced in it along a smooth horizontal surface.
• $InertialMass=\frac{AppliedForce}{AccelerationProduced}$
• $m_i$=F/a
• It is a measure of its ability to resist the production of acceleration by an external force.

Note: If some force is applied on two different bodies, then the inertial mass of that body will be more in which the acceleration produced is less and vice-versa.

3.0Measurement of Inertial Mass(Inertial Balance)

• The measurement of inertial mass of  a body is measured by using a device called inertial balance.
• It consists of a long strip of metal, one end of the strip is clamped to a table. The other end of the strip carries a pan in which a body whose inertial mass is to be measured is kept. 
• When the strip vibrates horizontally, its inertia comes into play and not the force of gravity of Earth.
• It is found that the period of Vibration T is directly proportional to the square root of the inertial mass m of the body.

$T\propto \sqrt{m}$ or $T^2\propto m$

• Let m1and m2 be the inertial masses of two objects and T1 and T2 be their corresponding periods of vibration, then $\frac{m_2}{m_1}=\frac{r_2^2}{r_1^2}$

Where $m_1$ is a standard mass ,unknown mass m2can be determined.

2. Gravitational Mass

• It determines the gravitational force acting on it due to the Earth is referred to as gravitational mass.
• It is defined by Newton’s Law of Gravitation. According to this law, the force of gravitation of the Earth on a body of mass $m_g$ is given by,

$F=G\frac{M{m}_g}{R^2}$

$m_g=\frac{F{R}^2}{GM},mg$ is the gravitational mass of the body which can be measured by using a physical balance.

•  The greater the gravitational mass of a body, the stronger the gravitational pull exerted on it by the Earth. Therefore, if two bodies at the same height experience equal gravitational force, their gravitational masses must be equal. This principle is the foundation of the pan balance.

4.0Measurement of Gravitational Mass(Physical Balance)

• A physical balance operates based on the principle of moments. When an object is in equilibrium under multiple forces acting in the same plane, the total clockwise moments are equal to the total anticlockwise moments.
• To weigh an object, it is placed in the left pan, while standard weights are placed in the right pan. The weights are adjusted until the beam becomes horizontal. At this point, the gravitational force on the object matches that of the standard weights, meaning the gravitational mass of the object is equal to that of the standard weights, effectively canceling the influence of gravitational acceleration (g).

5.0Equivalence of Inertial And Gravitational Mass

• According to The Newton’s Law of Gravitation, the gravitational force acting on a body of gravitational mass mg placed on the surface of the Earth is given by

$F=\frac{GM{m}_g}{R^2}$……..(1)

If a body of inertial mass miis allowed to fall freely then from Newton’s Second Law,

$F=\text{Inertial Mass}\times \text{acceleration due to gravity}=m_{i}g$ ………..(2)

From above two equations

$m_{i}g=\frac{GM{m}_g}{R^2}$

$\frac{m_i}{m_g}=\frac{GM}{g{R}^2}=k$(a Constant)

$m_i=k{m}_g$ or $m_i\propto m_g$

Note: The inertial mass of a body is proportional to its gravitational mass, value of k=1 so that$m_i=m_g$, inertial and gravitational masses are equivalent. Hence the force F=ma and weight equation w=mg ,we need to consider only one type of mass m.

6.0Difference between Inertial And Gravitational Mass

7.0Definition of Weight

• It is the measure of the gravitational pull exerted by the Earth upon it.
• The gravitational force with which a body is attracted towards the centre of the Earth.
• W=mg If the g is the acceleration due to gravity at a place then a body of mass m is attracted towards the centre of the Earth with a force equal to mg at that place.
• In vector Notation, W=mg Weight is a vector quantity.
• It is measured in the units of force such as Newton, kgwt etc.

Note: As the value of g varies from place to place the weight of a body also varies from place to place.

8.0Difference Between Mass And Weight