Mass and Weight

What is called Mass of a body?

Mass is a quantitative measure of matter content of a body which produce inertia. It is a fundamental property of all matter bodies. It offer a resistance in the body to a change in its speed or position upon the application of a force.

Therefore, mass of a body is defined as the amount of matter content in that body.

Mass of a body doesn’t change with the change in the position or location of body on the earth’s surface. It only changes when a part of body is cut out from it or another part is added to it.
The greater the mass of a body, the greater will be inertia and smaller will be the change produced by an applied force.
Mass of a body is measured by comparing it with a standard mass by using a lever balance. It is also known as a common balance.

What is called Weight of a body?

All material bodies are constructed of numerous tiny particles. These tiny particles of a body have their own masses. Hence, the total mass of the body is equal to the sum of all tiny masses by which the body is made of.

By gravitational nature of earth, each of these tiny particles is attracted towards the centre of the earth with a gravitational force, which is proportional to its mass. This is called weight of tiny particles. The sum of all the weights of tiny particles of a body, is called weight of that body.

Therefore, weight of a body is defined as the gravitational force of attraction by which earth pulls that body towards its centre.

Weight of a body depends upon earth’s acceleration due to gravity.
Hence, weight is called gravitational unit of force.
Earth’s acceleration due to gravity varies with the distance of body from centre of earth and also varies with different locations on earth’s surface.
Hence, weight of a body change with location on the earth’s surface.
Weight of a body is measured by using a spring balance.

Absolute unit of Force

A force produces or tends to produce a change in the state of rest or motion of a body. Therefore, a force brings a change in linear momentum of a body.

As per Newton’s second law of motion, the applied force on a body is directly proportional to the rate of change in momentum.

Therefore, $\quad P \propto \left (\text {Mass} \times \text {Acceleration} \right )$

If a force $( P )$ acting on a body of mass $( m )$ produces an acceleration or deceleration of $( a )$ in the body, then –

$P = k.m.a$

Here, $( k )$ is the constant of proportionality.

For convenience, the unit of force is so adopted such that, one unit of force produces one unit of acceleration to a body of one unit mass.

Therefore, when mass $( m = 1 )$ and acceleration $( a = 1 )$ , then we get force $( P = 1 )$ .

Thus, $\quad k = 1$

Putting this value, we have –

$P = m.a = \text {Mass} \times \text {Acceleration}$

The unit of force in $SI$ and $MKS$ systems is newton.

One newton $( 1 \ N )$ is that force which produces unit acceleration $( \alpha = 1 \ \frac {m}{s^2} )$ in a body of unit mass $( m = 1 \ Kg )$ .

Or, $\quad 1 \ N = 1 \ kg \times 1 \left ( \frac{ m }{ s^2 } \right ) = 1 \left ( \frac { Kg \ m }{ s^2 } \right )$

Gravitational unit of Force

In most of engineering purposes, a force is expressed in gravitational units.

We know that, absolute unit of force is newton. One newton $( 1 \ N )$ is that force which produces an acceleration of $( 1 \frac {m}{s^{2}} )$ in a body of mass $( 1 \ kg )$ .

Or, $\quad 1 \ N = 1 kg \times 1 \left ( \frac { m }{ s^2 } \right ) = 1 \left ( \frac { kg \ m }{ s^2 } \right )$

If same body is moving with an acceleration of $( 9.81 \frac {m}{s^2} )$ , then force acting on it will be $[ 1 \times 9.81 = 9.81 \ N ]$

Since, acceleration due to gravity of earth is $( g = 9.81 \frac {m}{s^2} )$ , so a body of $( 1 \ kg )$ mass is attracted towards the centre of earth with an acceleration of $( 9.81 \frac {m}{s^2} )$

In this case, the earth’s gravitational pull is called a force of $( 1 \text {kg-force} )$ or $( 1 \text {kgf} )$ .

But, earth’s gravitational pull on a body is called weight of that body.

Hence, force of $( 1 \text {kg-f} )$ is also represented as $( 1 \text {kg-wt} )$ .

Therefore, $1$ gravitational unit of force is equivalent to $9.81$ absolute units of force.

Gravitational unit of force i.e., $( 1 \text {kg-f} \ \text {or} \ 1 \text {kg-wt} )$ are engineer’s unit of force whereas, absolute unit of force i.e., $( 1 \ N )$ is a scientific unit or $SI$ unit of force.

TO BE NOTED –

In Absolute unit of force, mass $( \text {m} )$ of a body is considered.
In Gravitational unit of force, weight $( \text {mg} )$ of a body is considered.

Relation between Mass & Weight

Consider about a body of mass $( m = 100 \ kg )$ . Earth’s pulling force on this body will be –

$P = \text {Mass} \times \text {Acceleration}$

Or, $\quad P = 100 \times 9.81 = 981 \ N$

Because, $\quad \left [ g = 9.81 \left ( \frac {m}{s^2} \right ) \right ]$

Therefore, by definition of weight of a body –

$W = 981 \ N$

Therefore, $\quad W = \left ( \frac {981}{9.81} \right ) Kgf = 100 \ Kgf$

Because, $\quad ( 1 \ Kgf = 9.81 \ N )$ (Gravitational unit of force)