Mass and Momentum

Introduction

Mass of the body is the total quantity of matter present inside it. We know that matter is made up of atoms and molecules. Therefore, when we say that mass of the body is ‘n Kg’, it means that the total of individual masses of all the atoms and molecules present inside the body equals ‘n Kg’. The momentum and mass of a body are related to each other, provided that the body is in motion. Momentum is the measure of energy possessed by the moving body.

What is Mass?

The total amount of matter present inside a physical body is called it's Mass. Mathematically, Mass can be written as -

$$\mathrm{m\:=\:\rho\:V}$$

Where

𝑚 denotes the mass of the body.

𝜌 denotes the density of the body.

𝑉 denotes the volume of the body.

We all know that it is easy for us to push an empty carton over a carton full of heavy objects. Scientifically, the greater the mass of the body, the larger will be the amount of force required to push it. Therefore, the mass of the body also tells us about its ability to resist any change in its present state (either the state of rest or state of motion) against the net force applied to it. It should be kept in mind that mass is a Scalar quantity (it only has magnitude and no direction). The SI unit is Kg (Kilogram).

What is Weight?

Weight is the measure of the amount of force exerted by gravity on any object. Mathematically, weight can be written as

$$\mathrm{W\:=\:mg}$$

Here 𝑚 denotes the mass of the body and 𝑔 denotes the acceleration due to gravity Following things should be kept in mind about weight

• It is a vector quantity (it has both magnitude and direction- towards the center of the earth).

• Its SI unit is Newton

Basis of Comparison	Mass	Weight
Definition	Mass is the measure of inertia of the body.	Weight is the measure of the force exerted by gravity on the body.
Variation with location	It remains constant.	It changes with the value of acceleration due to gravity.

What is Momentum?

Consider a body of mass m moving with the velocity v on a straight line (1-D Motion). Then, the linear momentum of the body can be calculated using

$$\mathrm{p\:=\:mv}$$

It is a vector quantity. In vector form, it can be written as 𝑝⃗ = 𝑚⃗𝑣⃗. If we represent velocity in terms of its components along the three axes then

$$\mathrm{\overrightarrow{v}\:=\:v_{x}a_{x}\:+\:v_{y}a_{y}\:+\:v_{z}a_{z}}$$

where 𝑎𝑥 denotes the unit vector along 𝑥 − 𝑎𝑥𝑖𝑠, 𝑎𝑦 denotes the unit vector along

𝑦 − 𝑎𝑥𝑖𝑠, and 𝑎𝑧 denotes the unit vector along 𝑧 − 𝑎𝑥𝑖𝑠.

In such case, the momentum of a body in terms of its components along the three axes is given by$\mathrm{\overrightarrow{p}\:=\:mv_{x}\:+\:mv_{y}\:+\:mv_{z}}$ Where 𝑚𝑣𝑥, 𝑚𝑣𝑦 𝑎𝑛𝑑 𝑚𝑣𝑧 denotes the magnitude of momentum along x, y, and z-axis

The representation of these components is also shown in the diagram that follows

Law of Conservation of Linear Momentum

According to the law of conservation of momentum-If the net external force acting on a system is zero, then its momentum remains conserved ”.

To be more precise - The total momentum of the isolated system before and after any event remains constant ”

Let us understand this with the help of an example

Suppose that two cars A and B each of mass 𝑚 and 𝑀 respectively are moving towards each other with velocities 𝑣1 and 𝑣2. After some time, both cars collided and their velocities become 𝑣3 and 𝑣4 respectively. Now, for an isolated system, according to the law of conservation of momentum -

$$\mathrm{momentum\:before\:collision\:=\:momentum\:after\:the\:collision}$$

$$\mathrm{mv_{1}\:+\:mv_{2}\:=\:mv_{3}+\:mv_{4}}$$

However, it should be kept in mind that this law holds only for an isolated system - a system on which the net external force is 0.

Variation of Momentum with the Mass of the body

We all know that the linear momentum of a body of mass ‘m’ moving with velocity ‘v’ is given by

$$\mathrm{p\:=\:mv}$$

From the above equation it is clear that - Momentum of any moving body is directly proportional to its mass, keeping the velocity constant. But what is the significance of the momentum of any moving body? Let us understand this with the help of the following -

Case 1 − Suppose a car of mass 100 Kg is moving with the velocity of 50 m/s strikes a wall and came to rest.

Case 2 − Suppose a truck of mass 500 Kg is moving with the same velocity as that of the car at 50 m/s strikes a wall and came to rest.

Now, which vehicle - car or truck, provided more damage to the wall? This question can be easily answered if we can calculate and compare the momentum of both car and truck.

For Car − $$\mathrm{p_{1}\:=\:100\:\times\:50\:=\:5000kgms^{-1}}$$

For truck − $$\mathrm{p_{2}\:=\:500\:\times\:50\:=\:25000kgms^{-1}}$$

It is clear that the momentum of the truck is five times the momentum of the car. Momentum is also the measure of the amount of power a moving body possesses. In the above case, the damage imparted by the truck to the wall is five times the damage imparted by the car.

Conclusion

The mass of a body is the amount of matter present inside it. Mathematical physics is of no use if the mass of the body under observation is not known. The momentum of a body in motion is a measure of the power it possesses or the amount of energy it is moving with. Momentum Conservation is an important law and is used in a variety of real-life situations.

FAQs

1. Does the mass of the moving body always remain constant?

The mass of any moving body remains constant at normal velocities. However, the mass of the body starts to decrease as its speed reaches the speed of light in accordance with the special theory of relativity.

2. Give a real life example of conservation of momentum.

Recoil of the gun.

3. What does the rate of change of momentum tell us?

We know $$\mathrm{p\:=\:mv}$$

The rate of change of momentum will give $\mathrm{m\:\:\frac{dv}{dt}\:=\:ma\:=\:F}$ Hence, the rate of change of momentum gives the net force acting on the body.

4. is the relation between angular and linear momentum?

The relation between angular and linear momentum is given by -

$$\mathrm{L\:=\:rp\sin\theta}$$

5. What do you mean by Centre of Mass?

The Center of mass is a point in a body or system of bodies at which the whole mass of the body is assumed to be concentrated.