Work, Energy, Power and Collision

Introduction

The terms 'work', 'energy' and 'power' are frequently used in everyday language. A farmer clearing weeds in his field is said to be working hard. A woman carrying water from a well to her house is said to be working. In a drought affected region she may be required to carry it over large distances. If she can do so, she is said to have a large stamina or energy. Energy is thus the capacity to do work. The term power is usually associated with speed. In karate, a powerful punch is one delivered at great speed. In physics we shall define these terms very precisely. We shall find that there is a loose correlation between the physical definitions and the physiological pictures these terms generate in our minds.

Work is said to be done when a force applied on the body displaces the body through a certain distance in the direction of force.

WORK , TYPES OF WORK DONE , FRAME OF REFERENCE and GRAPHS

Work Done by a Constant Force

Let a constant force \(\overrightarrow{F}\) be applied on the body such that it makes an angle \(\theta\)  with the horizontal and body is displaced through a distance s

By resolving force \(\overrightarrow{F}\) into two components :

(i) F cos\(\theta\)  in the direction of displacement of the body.

(ii) F sin\(\theta\)  in the perpendicular direction of displacement of the body.

Since body is being displaced in the direction of \(F\cos \theta\), therefore work done by the force in displacing the body through a distance s is given by

\(W=(F\cos \theta )\,s=Fs\cos \theta\) or \(W=\overrightarrow{F}.\overrightarrow{s\,}\)

Thus work done by a force is equal to the scalar (or dot product) of the force and the displacement of the body.

If a number of forces \({{\overrightarrow{F}}_{1}},\,{{\overrightarrow{F}}_{2}},\,{{\overrightarrow{F}}_{3}}......{{\overrightarrow{F}}_{n}}\) are acting on a body and it shifts from position vector \({{\overrightarrow{\,r}}_{1}}\) to position vector \({{\overrightarrow{\,r\,}}_{2}}\) then \(W=({{\overrightarrow{F}}_{1}}+{{\overrightarrow{F}}_{2}}+{{\overrightarrow{F}}_{3}}+....{{\overrightarrow{F}}_{n}})\,.({{\overrightarrow{\,r}}_{2}}-{{\overrightarrow{\,r}}_{1}})\)

Nature of Work Done

Positive work

Positive work means that force (or its component) is parallel to displacement

\({{0}^{o}}\le \theta <{{90}^{o}}\)

The positive work signifies that the external force favours the motion of the body.

Example: (i) When a person lifts a body from the ground, the work done by the (upward) lifting force is positive.

(ii) When a lawn roller is pulled by applying a force along the handle at an acute angle, work done by the applied force is positive.

(iii) When a spring is stretched, work done by the external (stretching) force is positive.

Maximum work : \({{W}_{\max }}=F\,s\)

When  \(\cos \theta =\text{maximum}=1\)  i.e. \(\theta ={{0}^{o}}\)

It means force does maximum work when angle between force and displacement is zero.

Negative work

Negative work means that force (or its component) is opposite to displacement i.e.

\({{90}^{o}}<\theta \le {{180}^{o}}\)

The negative work signifies that the external force opposes the motion of the body.

Example: (i) When a person lifts a body from the ground, the work done by the (downward) force of gravity is negative.

(ii) When a body is made to slide over a rough surface, the work done by the frictional force is negative.

Minimum work : \({{W}_{\min }}=-F\,s\)

When \(\cos \theta =\text{minimum}=-1\) i.e \(\theta ={{180}^{o}}\)

It means force does minimum [maximum negative] work when angle between force and displacement is 180^o.

(iii) When a positive charge is moved towards another positive charge. The work done by electrostatic force between them is negative.

Zero work:

Under three condition, work done becomes zero \(W=Fs\cos \theta =0\)

(1) If the force is perpendicular to the displacement \(\mathbf{[}\overrightarrow{F}\,\bot \,\overrightarrow{s\,}\mathbf{]}\)

Example: 

(i) When a coolie travels on a horizontal platform with a load on his head, work done against gravity by the coolie is zero.

(ii) When a body moves in a circle the work done by the centripetal force is always zero.

(iii) In case of motion of a charged particle in a magnetic field as force \([\overrightarrow{F}=q(\overrightarrow{v\,}\times \overrightarrow{B})]\) is always perpendicular to motion, work done by this force is always zero.

(2) If there is no displacement [s = 0]

Example:

(i) When a person tries to displace a wall or heavy stone by applying a force and it does not move, then work done is zero.

(ii) A weight lifter does work in lifting the weight off the ground but does not work in holding it up.

Work Done by a Variable Force

When the magnitude and direction of a force varies with position, the work done by such a force for an infinitesimal displacement is given by \(dW=\overrightarrow{F}.\,d\overrightarrow{s\,}\)

The total work done in going from A to B as shown in the figure is

\(W=\int_{A}^{B}{\,\overrightarrow{F}.\,d\overrightarrow{s\,}=\int_{A}^{B}{\,(F\cos \theta )ds}}\)

In terms of rectangular component \(\overrightarrow{F}={{F}_{x}}\hat{i}+{{F}_{y}}\hat{j}+{{F}_{z}}\hat{k}\)

\(d\overrightarrow{s\,}=dx\hat{i}+dy\hat{j}+dz\hat{k}\)

\(\therefore \,\,\,W=\int_{A}^{B}{({{F}_{x}}\hat{i}+{{F}_{y}}\hat{j}+{{F}_{z}}\hat{k})}.(dx\hat{i}+dy\hat{j}+dz\hat{k})\)

or \(W=\int_{{{x}_{A}}}^{{{x}_{B}}}{{{F}_{x}}dx+\int_{{{y}_{A}}}^{{{y}_{B}}}{{{F}_{y}}dy}}+\int_{{{z}_{A}}}^{{{z}_{B}}}{{{F}_{z}}dz}\)

Dimension and Units of Work

Dimension :  As work =  Force \(\times\) displacement

[W]\(=[ML{{T}^{-2}}]\times [L]=[M{{L}^{2}}{{T}^{-2}}]\)

Units :  The units of work are of two types

Let a body, whose initial position is \({{x}_{i}}\), is acted upon by a variable force (whose magnitude is changing continuously) and consequently the body acquires its final position \({{x}_{f}}\).

Let F be the average value of variable force within the interval dx from position x to (x + dx) i.e. for small displacement dx. The work done will be the area of the shaded strip of width dx. The work done on the body in displacing it from position \({{x}_{i}}\) to \({{x}_{f}}\) will be equal to the sum of areas of all the such strips

\(dW=\overrightarrow{F}\,dx\)

\(\therefore \,W=\int_{{{x}_{i}}}^{{{x}_{f}}}{dW=\int_{{{x}_{i}}}^{{{x}_{f}}}{F\,dx}}\)

\(\therefore \,W=\int_{{{x}_{i}}}^{{{x}_{f}}}{(\text{Area}\,\text{of}\,\text{strip}\,\text{of}\,\text{width}\,dx)}\)

\(\therefore \,W=\text{Area}\,\text{under}\,\text{curve}\,\text{between}\,{{x}_{i}}\,\text{and}\,{{x}_{f}} \)   

Work Done in Conservative and Non-conservative Field

(1) In conservative field, work done by the force (line integral of the force i.e. \(\int{\overrightarrow{F}.d\overrightarrow{l\,}}\)) is independent of the path followed between any two points

\(\underset{\begin{smallmatrix} \, \\ \text{Path I} \end{smallmatrix}}{\mathop{{{W}_{A\to B}}}}\,=\underset{\begin{smallmatrix} \, \\ \text{Path II} \end{smallmatrix}}{\mathop{{{W}_{A\to B}}}}\,=\underset{\begin{smallmatrix} \, \\ \text{Path III} \end{smallmatrix}}{\mathop{{{W}_{A\to B}}}}\,\,\,or \,\,\\ \,\,\underset{\begin{smallmatrix} \, \\ \text{Path I} \end{smallmatrix}}{\mathop{\int{\overrightarrow{F}.d\overrightarrow{l\,}}}}\,=\underset{\begin{smallmatrix} \, \\ \text{Path II} \end{smallmatrix}}{\mathop{\int{\overrightarrow{F}.d\overrightarrow{l\,}}}}\,=\underset{\begin{smallmatrix} \, \\ \text{Path III} \end{smallmatrix}}{\mathop{\int{\overrightarrow{F}.d\overrightarrow{l\,}}}}\,\)

(2) In conservative field work done by the force (line integral of the force i.e. \(\int{\overrightarrow{F}.d\overrightarrow{l\,}}\)) over a closed path/loop is zero.

\({{W}_{A\to B}}+{{W}_{B\to A}}=0\)

or  \(\oint{\vec{F}.d\vec{l}=0}\)

Conservative force : The forces of these type of fields are known as conservative forces.

Example : Electrostatic forces, gravitational forces, elastic forces, magnetic forces etc and all the central forces are conservative in nature.

If a body of mass m lifted to height h from the ground level by different path as shown in the figure

Work done through different paths

\({{W}_{I}}=F.\,s=mg\times h=mgh\)

\({{W}_{II}}=F.\,s=mg\sin \theta \times l=mg\sin \theta \times \frac{h}{\sin \theta }=mgh\)

\({{W}_{III}}=mg{{h}_{1}}+0+mg{{h}_{2}}+0+mg{{h}_{3}}+0+mg{{h}_{4}}\)

\(=mg({{h}_{1}}+{{h}_{2}}+{{h}_{3}}+{{h}_{4}})=mgh\)

\({{W}_{IV}}=\int{\overrightarrow{F}.\,d\overrightarrow{s\,}}=mgh\)

It is clear that \({{W}_{I}}={{W}_{II}}={{W}_{III}}={{W}_{IV}}=mgh\).

Further if the body is brought back to its initial position A, similar amount of work (energy) is released from the system, it means \({{W}_{AB}}=mgh\) and \({{W}_{BA}}=-mgh\).

Hence the net work done against gravity over a round trip is zero.

\({{W}_{Net}}={{W}_{AB}}+{{W}_{BA}}=mgh+(-mgh)=0\)

i.e. the gravitational force is conservative in nature.

Non-conservative forces :

A force is said to be non-conservative if work done by or against the force in moving a body from one position to another, depends on the path followed between these two positions and for complete cycle this work done can never be zero.

Example:  Frictional force, Viscous force, Airdrag etc.

If a body is moved from position A to another position B on a rough table, work done against frictional force shall depend on the length of the path between A and B and not only on the position A and B.

\({{W}_{AB}}=\mu mgs\)

Further if the body is brought back to its initial position A, work has to be done against the frictional force, which opposes the motion. Hence the net work done against the friction over a round trip is not zero.

\({{W}_{BA}}=\mu mgs.\)

\(\therefore \,{{W}_{Net}}={{W}_{AB}}+{{W}_{BA}}=\mu mgs+\mu mgs=2\mu mgs\ne 0.\)

i.e. the friction is a non-conservative force.

Work Depends on Frame of Reference

With change of frame of reference (inertial), force does not change while displacement may change. So the work done by a force will be different in different frames.

Examples : (1) If a porter with a suitcase on his head moves up a staircase, work done by the upward lifting force relative to him will be zero .(as displacement relative to him is zero) while relative to a person on the ground will be mgh.

(2) If a person is pushing a box inside a moving train, the work done in the frame of train will \(\overrightarrow{F}.\overrightarrow{s\,}\) while in the frame of earth will be \(\overrightarrow{F}.\,(\overrightarrow{s\,}+{{\overrightarrow{\,s}}_{0}})\) where \({{\overrightarrow{\,s}}_{0}}\) is the displacement of the train relative to the ground.