Newton’s laws of motion and applications


This article is about Newton’s laws of motion and applications for B.Sc first year students and is also applicable if you are preparing for exams like JAM , GATE, JEST , CSIR-NET etc. It comes under the subject Mechanics. If you like the content do help us by sharing it among your fellow students.

Introduction

Sir Isaac Newton expressed his ideas regarding the motion of bodies in the form of three laws which are considered as the basic laws of mechanics. In fact Mechanics is the study of certain general relations that describe the interactions of material bodies and the Laws governing them. One general property of a material body is its inertial mass. The concept of describing interactions between material bodies if ‘force’. These two concepts of inertial mass and force were first defined in a quantitative manner by Isaac Newton.
Newton’s Laws of motion are of utmost importance in the field of physics.

It is important to understand which part of Newton’s laws are based on experimet and which parts are matter of definitions. While studying these laws we must learn to apply them as their application is essential for real understanding of underlying concepts. (An Introduction to Mechanics )

Newton’s Laws gives us the direct introduction of classical mechanics but it is not the only theory to do so. There are the formulations of Lagrange and Hamilton which take energy instead of force as the fundamental concept. As for the results are concerned these methods are equivalent to Newtonian approach and understanding of Newtonian mechanics is helpful in understanding other treatments of mechanics.

As such Newton’s laws of motion forms the core of classical dynamics. These laws are explained as follows.

Newton’s First Law of motion

Everybody continues to be in its state of rest or of uniform motion in a straight line unless acted upon by an external unbalanced force

Newton’s first law of motion talk about two things:-

(a) force (or its absence): which is an expression of the interaction between different objects.

(b) measurement of rest or uniform motion of the body or object

This law of states that body would either remain at rest or continue to move with constant velocity. But the question arises “with respect to whom”? For this, we need a frame of reference.

Now consider a situation, where the observer is sitting in an accelerated frame ( velocity of the frame changing with time). Then, in this case, the object can not always remain at rest or in uniform motion w.r.t. the observer. A reference frame w.r.t. which first law holds is an inertial frame of reference. So, not all coordinate systems or frame of reference are inertial. It is always possible to find a coordinate system with respect to which isolated bodies move uniformly. This is the essence of Newton’s first law of motion. So this law of motion is the assertion that the inertial system exists. An inertial system must be a frame moving with a constant velocity (non accelerating). The question whether a given frame of reference is inertial or not is a matter of observation and experiment.

Note:- For most of the observations made on the surface of the earth, the frame of reference attached to the surface of Earth behaves as an inertial frame to a very good approximation.

Newton’s Second Law of motion

The rate of change of linear momentum is proportional to the impressed force and takes place in the direction of the applied force.

We can also say that, the time rate of change of linear momentum of an object is proportional to the external force acting on it and occurs in the direction of the resultant force. That is

$\vec F = k\frac{{d\vec p}}{{dt}} = km\vec a$

Here $\vec{F}$ is the resultant external force, m is the inertial mass of the object, and $\vec p = m\vec v$ is the linear momentum of the object. K is the constant of proportionality which could be taken as 1 by defining appropriate units.

Here let us be careful about a point that Newton’s second law is not a definition of force. The equation $\vec F = m\vec a$ is an equality which says that the force acting on a body determines the rate of change of velocity. What is the nature and value of $\vec{F}$  is unanswered in in second law.

Newton’s second law of motion is valid for an object having constant inertial mass and moving with velocity small compared to that of light. If the velocity of the object is significantly high (i.e., a significant fraction of the velocity of light), we enter into the domain of special theory of relativity and Newton’s laws were revised in relativistic dynamics.

Newton’s Third Law of motion

When two objects interact, the mutual force of interaction on each object is equal in magnitude and opposite in direction. That is ‘to every action there is an equal and opposite reaction’.

In the third law of motion, the action means the force due to the body (1) on the body (2) and may be expressed by ${\vec F_{12}}$. The reaction here means the force due to the body (2) on the body (1) and may be expressed as ${\vec F_{21}}$. The third law can be expressed as

$\vec{F}_{12}=-\vec{F}_{21}$

There is no such thing as lone force without a partner. Newton’s third law is the statement about the nature of the force. Third law of motion leads directly to the law of conservation of momentum.

Newton’s third law of motion implies that the forces exerted by the two interacting bodies over each other are equal and opposite, provided both the forces are measured simultaneously. A simultaneous measurement of the two forces is possible for ordinary practical purposes if the time taken by the interaction is sufficiently large compared with the time taken by light signal to travel one to the other. This means that the law ceases to hold good for particles of atomic dimensions, for which simultaneous measurement of two forces is almost impossible.

Solved Example 1. Show that  Newton’s first law of motion is simply a special case of Newton’s second law.

Solution. Newton’s second law of motion gives a measure of a force as the rate of change of linear momentum. If $\vec{p}$ is the momentum of the particle, given by, where m is mass and $\vec{p}$ its velocity, then

$\vec{F}=\frac{d\vec{p}}{dt}=\frac{d}{dt}\left( m\vec{v} \right)$

$\vec{F}=m\frac{d\vec{v}}{dt}=m\vec{a}$     (m being constant)

so that, if $\vec{F}=0$,  then, $\vec{a}=0$ which is the first law of motion, i.e. if no force acts, there will be no change in velocity and hence no acceleration. Thus, Newton’s first law of motion is simply a special case of the second law.

Conservation of linear momentum

According to Newton’s first law of motion, ‘a body remains at rest or continues to move at a uniform velocity, as long as no external force is acting on it.’ According to Newton’s second law of motion, ‘the rate of change of linear momentum of a body is proportional to the force acting on it.’

Using Newton’s first law of motion, if a particle of mass m is moving with velocity $\vec{v}$, its linear momentum is $p=m\vrc{v}$. When a force $\vec{F}$  is applied to it, its momentum changes at the rate $\frac{d\vec{p}}{dt}$.



$\vec{F}=\frac{d\vec{v}}{dt}=\frac{d(m\vec{v})}{dt}$

Clearly when

$\vec{F}=0,\frac{d\vec{p}}{dt}=0$

Or.

$\vec{p}=m\vec{v}$= a constant

i.e.,  if no force acts, the linear momentum is conserved. Thus Newton’s first law of motion is only a special case of second law.

Using  Newton’s second law of motion, now consider a system of two particles which are acted only by forces of interaction like Coulomb’s forces or gravitational forces and no external forces act on the system.

Let $\vec{F}_{12}$ be the force exerted by the first particle on the second, known as action and $\vec{F}_{21}$ be the force exerted by the second on the first, known as reaction, then according to Newton’s third law of motion, action and reaction being equal and opposite. That is,

$\vec{F}_{12}=-\vec{F}_{21}$

According to Newton’s second law of motion

${{\vec{F}}_{12}}=\frac{d{{{\vec{p}}}_{2}}}{dt}=\frac{d({{m}_{2}}{{{\vec{v}}}_{2}})}{dt}={{m}_{2}}\frac{d\vec{v}_2}{dt}$

where $m_2$, $\vec{v}_2$ and $\vec{p}_2$ are the mass, velocity and momentum of the second particle.

Similarly ,

${{\vec{F}}_{21}}=\frac{d{{{\vec{p}}}_{1}}}{dt}=\frac{d({{m}_{1}}{{{\vec{v}}}_{1}})}{dt}={{m}_{1}}\frac{d\vec{v}_1}{dt}$

where $m_1$, $\vec{v}_1$ and $\vec{p}_1$ are the mass, velocity and

$\vec{F}_{12}=-\vec{F}_{21}$

Or

${{\vec{F}}_{12}}+{{\vec{F}}_{21}}=0$.

So, we have

$ \frac{d{{{\vec{p}}}_{1}}}{dt}+\frac{d{{{\vec{p}}}_{2}}}{dt}=0 $

or,

$ \frac{d}{dt}\left( {{{\vec{p}}}_{1}}+{{{\vec{p}}}_{2}} \right)=0$

Integrating, we have

$\vec{p}_1+\vec{p}_2=$ a constant  or $m_1\vec{v}_1+m_2\vec{v}_2 =$ a constant

The quantity

$\vec{p}_1+\vec{p}_2= m_1\vec{v}_1+m_2\vec{v}_2 $

represents the total linear momentum of the system. Hence we conclude that ‘If Newton’s second and third law of motion holds good, the total linear momentum of the system of two particles remains constant’. The above law can be extended to a system of three or more interacting particles. The law of conservation of linear momentum is, therefore, a basic law and is stated as under. “The total linear momentum of a system of particles free from the action of external forces and subjected only to their mutual interaction remains constant, no matter how complicated the forces are.”


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