Some of the most important Physics Topics include energy, motion, and force.

## How to Find the Inertia of an Object

The mass of a body is the quantity of matter (or material) contained in it. Mass is a scalar quantity

which has only magnitude but no direction. The mass of a body (or object) is commonly measured by an equal arm balance (see Figure 30). The SI unit of mass is kilogram which is written in short form as kg. A

body contains the same quantity of matter wherever it be—whether on earth, moon or even in outer space. So, the mass of an object is the same everywhere. For example, if the mass of an object is 5 kilograms on the earth, then it will have the same mass of 5 kilograms even when it is taken to any other planet, or moon, or in outer space. Thus, the mass of a body (or object) is constant and does not change from place to place. Mass of a body is usually denoted by the small ‘m’. Mass of a body is a measure of inertia of the body and it is also known as inertial mass. The mass of a body cannot be zero.

### Weight

The earth attracts every body (or object) towards its centre with a certain force which depends on the mass of the body and the acceleration due to gravity at that place. The weight of a body is the force with which it is attracted towards the centre of the earth. In other words, the force of earth’s gravity acting on a body is known as its weight.

We know that, Force = mass × acceleration

The acceleration produced by the force of attraction of the earth is known as acceleration due to gravity and written as ‘g’. Thus, the downward force acting on a body of mass ‘m’ is given by :

Force = mass × acceleration due to gravity

or Force = m × g

But, by definition, the force of attraction of earth on a body is known as weight W of the body, so by writing weight W in place of force in the above equation, we get:

Weight, W = m × g

where m = mass of the body

and g = acceleration due to gravity

Weight is measured in the same units as force. We know that the SI unit of force is newton (N). So, the SI unit of weight is also newton which is denoted by the letter N. Let us calculate the weight of 1 kilogram mass. We know that :

Weight, W = m × g

= 1 kg × 9.8 m/s^{2}

= 9.8 × 1 kg × 1 m/s^{2}

Now, by definition 1 kg × 1 m/s^{2} is equal to 1 newton, so :

Weight, W = 9.8 newtons (or 9.8 N)

Thus, the weight of 1 kilogram mass is 9.8 newtons. This means that the force acting on a mass of 1 kilogram at the surface of the earth is 9.8 newtons.

We have just seen that : W = m × g. Now, at a given place, the value of g is constant, therefore, at a given place W ∝ m, that is, at a given place, the weight of a body is directly proportional to its mass. It is due to this reason that at a given place, we can use the weight of a body as a measure of mass of the body.

Weight is a vector quantity having magnitude as well as direction. The weight of a body acts in vertically downward direction. The weight of a body is usually denoted by W. The weight of a body is given by W = m × g, and since the value of g (the acceleration due to gravity) changes from place to place, therefore, the weight of a body also changes from place to place. Thus, the weight of a body is not constant. In the interplanetary space, where g = 0, the weight of a body becomes zero and we feel true weightlessness. Thus, the weight of a body can be zero.

We know that the value of acceleration due to gravity g, decreases as we go down inside the earth and becomes zero at the centre of the earth. So, whatever be the weight of a body on the surface of the earth, its weight becomes zero when it is taken to the centre of the earth (because the value of g is zero at the centre of the earth). The weight of an object is measured with a spring balance (see Figures). The

spring balance gives us the weight of the object because the extension of the spring depends on the force with which it is pulled downwards by the earth. Thus, it is the gravitational force acting on an object which operates a spring balance, and not its mass.

### Weight of an Object on the Moon

Just as the ‘weight of an object on the earth is the force with which the earth attracts the object, in the same way, the weight of an object on the moon is the force with which the moon attracts that object. The gravitational force of the moon is about one-sixth that of the earth, therefore, the weight of an object on the moon will be about one-sixth of what it is on the earth. Thus, a spring balance which shows the weight of a body to be 6 N on earth will show a weight of only 1 N when taken to the moon. The weight of an object on the moon is less than that on the earth because the mass and radius of moon are less than that of earth (due to which it exerts a lesser force of gravity).

To Show That the Weight of an Object on the Moon is \(\frac{1}{6}\)th of Its Weight on the Earth

Suppose we have an object of mass m. Let its weight on the moon be Wm. Again suppose that the mass of the moon is M and its radius is R. Now, according to universal law of gravitation, the weight (force) of the object on the moon will be :

Weight on moon, W_{m} = G × \(\frac{M \times m}{R^2}\) … (1)

This weight is actually the force with which the moon attracts the object.

Suppose the weight of the same object on the earth be W_{e}. Now, we know that the mass of the earth is 100 times that of the moon, and the radius of the earth is 4 times that of the moon. Thus, if mass of moon is M, then the mass of earth will be 100M; and if radius of moon be R, then the radius of earth will be 4R. Now, taking the mass of earth as 100M and radius of earth as 4R, and applying universal law of gravitation, the weight of object on the earth will be :

Weight on earth, W_{e} = G × \(\frac{100 M \times m}{(4 R)^2}\)

or W_{e} = G × \(\frac{100 M \times m}{16 R^2}\) …. (2)

Now, dividing equation (1) by equation (2), we get:

\(\frac{W_m}{W_e}\) = \(\frac{G \times M \times m \times 16 R^2}{R^2 \times G \times 100 M \times m}\)

or \(\frac{W_m}{W_e}\) = \(\frac{16}{100}\)

or \(\frac{W_m}{W_e}\) = \(\frac{1}{6}\)

or \(\quad \frac{\text { Weight on moon }}{\text { Weight on earth }}\) = \(\frac{1}{6}\)

or weight of moon = \(\frac{1}{6}\) weight of earth

Please note that the mass of the object is the same on the moon and the earth, but its weight on the

moon is only one-sixth(\(\frac{1}{6}\)) of the weight on the earth. We will now solve some problems based on mass and weight. Please note that if the value of acceleration due to gravity of earth, g, is not given in the numerical problems, then we should take its value to be 9.8 m/s^{2}.

**Example Problem 1.**

Mass of a body is 5 kg. What is its weight ?

**Solution.**

The weight of a body is calculated by using the formula :

Weight, W = m × g

Here, Mass of the body, m= 5 kg

And, Acceleration due to gravity, g = 9.8 m/s^{2}

So, Weight, W = 5 × 9.8

= 49 N

Thus, the weight of the body is 49 newtons.

**Example Problem 2.**

What is the mass of an object whose weight is 49 newtons ?

**Solution.**

Here, Weight, W = 49 N

Mass, m = ? (To be calculated)

And, Acceleration due to gravity, g = 9.8 m/s^{2}

Now, putting these values in the formula,

W = m × g

we get : 49 = m × 9.8

m = \(\frac{49}{9.8}\)

m = 5 kg

Thus, the mass of the object is 5 kilograms.

**Example Problem 3.**

A man weighs 600 N on the earth. What is his mass ? (take g = 10 m s^{-2}). If he were taken to the moon, his weight would be 100 N. What is his mass on the moon ? What is the acceleration due to gravity on the moon ?

**Solution.**

First of all we will find out the mass of man on the earth. This can be done by using the formula:

W = m × g

we get : 49 = m × 9.8

m = \(\frac{49}{9.8}\)

m = 5 kg

Thus, the mass of object is 5 kilograms.

**Example Problem 3.**

A man weighs 600 N on the earth. What is his mass? (take g = 10 m 2). If he were taken to the moon, his weight would be 100 N. What is his mass on the moon? What is the acceleration due to gravity on the moon?

**Solution.**

First of all we will find out the mass of man on the earth. This can be done by using the

formula: W = m × g

Here, Weight of man on earth, W = 600 N

Mass of man on earth, m = ? (To be calculated)

And, Acceleration due to gravity(on earth), g = 10 ms^{-2}2

Now, putting these values in the above formula, we get: .

600 = m × 10

So, m = \(\frac{600}{10}\)

m = 60 kg

Thus, the mass of man on the earth is 60 kilograms. Now, the mass of a body remains the same everywhere in the universe. So, the mass of this man on the moon will also be 60 kilograms.

We will now calculate the value of acceleration due to gravity on the moon by using the same formula :

W = m × g

Now, Weight of man on the moon, W = 100 N

Mass of the man on moon, m = 60 kg (Calculated above)

And, Acceleration due to gravity(on the moon), g = ? (To be calculated)

By putting these values in the above formula, we get:

100 = 60 × g

So, g = \(\frac{100}{60}\)

g = 1.66 m s^{-2}

Thus, the acceleration due gravity on the surface of the moon is 1.66 m s^{-2}.

**Example Problem 4.**

How much would a 70 kg man weigh on the moon ? What would be his mass on the earth and on the moon ? (Acceleration due to gravity on moon = 1.63 m/s^{2})

**Solution.**

We will first calculate the weight of the man on the moon.

Here, Mass of the man on moon, m = 70 kg

Acceleration due to gravity(on the moon), g = 1.63 m/s^{2}

We know that: W = m × g

So, W= 70 × 1.63

W = 114.1 N

Thus, the man would weigh 114.1 newtons on the moon. Please note that the mass of a body is constant everywhere in the universe. So, the mass of this man would be the same on the earth as well as on the moon, that is, the mass will be 70 kg on the earth as well as on the moon.

Before we end this discussion on mass and weight, we would like to give the main differences between mass and weight in tabular form.

Differences Between Mass and Weight