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Work
Whenever a force displaces its point of application in a direction parallel to itself work is
done. Only a force can do work.
When work is done by a force, energy is transformed from one type to another.
e.g. 1:
When a body falls accelerating under gravity, the work done is the amount by which
the potential energy of the system drops and its kinetic energy rises.
e.g. 2:
When a body moves against a frictional force, the work done is the amount by which
sound and heat energy are produced at the expense of mechanical energy.
If the point of application of a constant force F is displaced by an amount s in a direction
making an angle with the force, then the work done is defined as:
Work is a scalar quantity.
When unit force displaces its point of application through unit distance in a direction parallel to
itself, unit work is done.
1 Joule
1 erg
=
=
1 newton
1 dyne
x
x
1 metre
1 centimetre
[MKS]
[cgs]
The joule and the erg are respectively the MKS and the cgs units of work. These units are
also called the absolute units of work as they are of constant value at all locations. Another set of
units of work - the gravitational units - are defined as the product of the gravitational unit of force
times the unit of distance.
A force does zero work if:



there is no displacement of its point of application (s = 0) e.g. No work is done by the force
of static friction since there is no displacement of its point of application
the point of application is displaced perpendicular to itself (cos =0)
e.g. There is no work done by the gravitational force (weight) when one walks on a
horizontal surface.
the
force
itself
is
of
negligible
magnitude
(F=0)
e.g. No work is done by intermolecular forces when a gas at low pressure undergoes
expansion.
Work Done from Force-Displacement graph
If the force remains constant over the displacement s directed parallel to itself, then the
work W done, is the product of F and s. This quantity is also given by the area enclosed under the
graph shown below.
If the force varies with respect to the displacement then the work done is still the area
enclosed under the graph.
The displacement s may be considered divided into a great number of infinitesimally small
sections ds such as the one shown in the figure alongside.
Over this tiny displacement the force acting may be considered to remain constant as
shown in figure (b). Thus the small amount of work dW done due to this amount of displacement is
given by the product of the constant force F and the parallel displacement s which is the area of
the corresponding rectangle.
Therefore,
dW = F ds
Over the entire displacement from s1 to s 2 the work W done is given by the sum of the
areas of all such rectangles between the limits of s1 and s2 as shown by the shaded area in figure
(c). The process of this summation is called integration.
Positive Work and Negative Work
The magnitude of work done is given by
The magnitude of the force vector cannot be negative. The magnitude of the displacement
vector likewise cannot be negative. Hence if the angle between the positive directions of these
two vectors is an obtuse angle, cos being negative, the work done by the force is also negative.
Hence whenever the component of force parallel to the displacement is oppositely directed to the
displacement, the work done becomes negative.
When a body is lifted, the lifting force does positive work since it is in the vertically upward
direction. The weight of the body being oppositely directed to the displacement, does negative
work. When a spring is stretched, the work done by the stretching force is positive while the work
done by the tension in the spring is negative. When a body slides along a rough surface the work
done by the force of kinetic friction is always negative since it is directed oppo-site to the
displacement.
Power
The rate of doing work is called power. Hence,
Unit power is unit work per unit time
1 Watt = 1 Joule/second
Power is a scalar quantity.
Energy and its Conservation
Energy of a body is the ability of the body to do work. The energy of a body or a system of bodies
is measured by the amount of work it can do.
The unit of energy is therefore the same as the unit of work.
Energy is a scalar quantity.
The law of conservation of energy states that in any closed system, the total energy is a constant,
but energy can be changed from one form to another.
It should be borne in mind that mass is also a form of energy just as are chemical energy thermal
energy nuclear energy etc.
In this chapter it is only mechanical energy that is the topic under study. Mechanical energy is of
two types:
a. kinetic energy and
b. potential energy
Kinetic Energy
Moving water has energy as it has the ability to move floating objects and itself.Still water
does not possess this ability. Moving air can move itself as well as kites, leaves, clothes etc. Still
air does not have this ability.
Hence all moving objects have energy which is specifically because of its motion.
Such energy which exists in a moving body because of its motion is called kinetic energy.
Let a body of mass m moving initially with a velocity u be acted upon by an opposing
force F which decelerates the body. Let the final velocity of the body be zero after a distance s is
covered.
Therefore, the work done by the body is
F . s = F x s cos 180o
=- Fs
Thus, energy possessed by the body is also equal to the same amount.
Thus kinetic energy of a body of mass m moving with a velocity v is given by
mv2
Potential Energy
A body placed at a height above the earth and released subsequently under-goes
downward displacement i.e. towards its position of rest. The free end of a rubber cord pulled taut
and then released undergoes displacement towards its position of rest. A simple pendulum
displaced from its position of rest subsequently undergoes displacement towards its rest position.
All these bodies undergo displacement because of their being placed away from their normal
resting position. All these bodies therefore posses energy by virtue of their displaced positions.
Such energy is called potential energy.
If a body has an external unbalanced force acting on it, only then can the body can move
itself if released. This external force is exerted upon the displaced body by another body which in
turn experiences an equal and opposite reaction force. Thus the two body system has potential
energy. A body may also have energy by virtue of the positions of its multiple parts with respect to
each other i.e. the energy is a property of a system of many objects. This is its potential energy by
virtue of its configuration.
The system comprising the apple and the earth has potential energy as there is relative
displacement (from the rest position) between its parts.
The system comprising cork, water and earth has potential energy by virtue of the position of its
various parts with respect to each other.
The system comprising the mass and the stretched spring has potential energy by virtue of
the position of its different parts with respect to each other.
The system comprising the two charged bodies has potential energy by virtue of the
position of its two parts with respect to each other.
If the displacement from rest position is oppositely directed to the system's force, then the
system's force does work by an amount
W = Fs s cos 180o = - FsS
This actually implies that an external force did positive work (displaced the body). The system thus
gains in potential energy by the amount of work done by the external force. Work done by external
force is thus given by:
= FE s cos 0o = + FES
[Since there is to be no gain in kinetic energy, there was no acceleration, and thus no
unbalance between the system's force and the external force acting. In magnitude, the external
force was equal to the system's force. The direction was opposite to the system's force and thus
the same as the displacement.]
If the displacement is in the direction of the system's force then the system's force does
positive work. the system loses this amount of potential energy. Increase in potential energy is the
work done by the external force and is given by the product of the external force (equal in
magnitude to the system's force) and the displacement parallel to the force.
If the system's force FS is constant in magnitude and direction over the entire parallel
displacement s (=s' cos ), then the required external force FE is also a constant.
Thus, gain in potential energy = FE.s
In case the force varies with respect to the displacement, then the work done is given by
where FE (= FS ) is a function of the displacements
[It is only in the case of a system comprising an object and the earth that the system's force acting
on the object is gravitational and equal to mg . It is only in this particular case that a small
displacement h in a vertical direction causes a change of the amount mgh in its potential energy.
In the event that the vertical displacement for this system is large enough for the acceleration due
to gravity to alter, the potential energy differs from mg and is to be found by integration as above
In case of the other systems discussed above the potential energy is not equal to mgh.]
Work-Energy relation
If there are opposing forces acting on a body then the work done ( W) by the net resultant
force
F on a particle is equal to its gain in kinetic energy.
That is, W = K
where,
K .= Rise in kinetic energy of the system
P .= Rise in potential energy of the system
U= Rise in thermal energy of the system
Energy of Freely Falling Body
Let a body of mass m initially stationary at a height h fall through a vertical distance x. Let
the acceleration due to gravity g remain constant through this small distance of fall.
Therefore, for a freely falling body as it loses potential energy it gains an equal amount of
kinetic energy if there is no other form of energy being produced (i.e. if there is no frictional force).
In case there is friction, then the body loses mechanical energy by the same amount as is the
energy dissipated as heat, sound etc.
Energy for the future
No discussion on energy is complete without considering the complex problem of energy
for the future. Such a discussion must incorporate the simple areas of technical knowledge and
economics as well as sensitive issues of politics and personal feelings regarding how much
damage to the environment man- kind is willing to tolerate.
The last named is in itself incurs volatile differences of opinion as yet un-solved. The USA
with approximately 5% of the world population consumes not less than 35% of the total energy
produced in the world.
The energy production crisis facing mankind is not merely due to the rising world polulation
along with the industrialization of the developing countries. The per capita increase of demand for
energy has climbed drastically in the last few decades and is expected to continue.
All technological studies show that there are only two sources that are truly capable of
meeting long range demands. The first of these is solar energy. This is virtually "inexhaustible" as
the estimated remaining lifetime of our sun is around 5 billion years.
The second promising source of energy sufficient to meet mankind's long range
requirements is the fusion of deuterium from sea water. Deuterium is an isotope of hydrogen.
When two nuclei of deuterium fuse to form one of helium, some mass m disappears in the
reaction to reappear as energy(E= m c2 ).
Although per 6500 atoms of hydrogen there is only one atom of deuterium (1H2) and the
amount of water in the oceans is finite, still this is not only an impressively large reserve of energy
but an inexpensive one as well. 1 km 3 of sea water has more energy content than the world's
entire fossil fuel reserves combined.
Fusion, as a potential source of energy for peaceable purposes is however, still to be
harnessed. There are as yet insurmountable technical problems. But assuming that technology
will overcome such problems in due course, how effective would be this enormous source be in
meeting the world's long range needs?
Fusion reactors would probably be used to produce electricity. If the world's consumption of
electricity of the year 1984 be used as an index of annual consumption, the deuterium in the
oceans would suffice as an energy source for the next 5 billion years - the estimated lifetime of our
sun. That is an impres-sive span of time. Then where is the energy crisis? We are quite
comfortably off with respect to this reserve.
Not so at all. Let us re-calculate the stock position keeping in mind the growing demand for
energy. To appreciate the dramatic and sobering proportions of this crisis, let us view just one
aspect of this rising demand - that of the geometric growth in demand. It is found that the growth
pattern of any measurable quantity initially shows a geometric or exponential characteristic. This is
true of bacterial growths, the number of visitors to national parks, automobile sales, sales of
trendsetter garments, and so on.
Assuming that a is the initial yearly consumption, r the annual rate of growth and n the total
number of years, we have,
Total available energy = a + a(1+r) + a(1+r)2 + a(1+r)3 + ...............a(1+r)(n-1)
Total available energy =
Substituting 3% as the annual growth rate in demand and solving for n, we find that the
available reserve will drop from the previous estimate of 5 billion years to under 650 years!!!!
Exponential growths however, cannot continue forever. They must come to a halt eventually as
they proceed to change the very balance favouring geometric growths. The calculation above,
though mathematically accurate, is absurd in the sense that other limits will have curbed this
growth rate long before 650 years have elapsed. For example with the current growth rate in
population (1.7% p.a.) in approximately 350 years there will be no more than one square metre of
ground space per person on earth. Unless this growth rate is checked, obviously disaster will
strike in the form of war, famine, pollution or pestillence cutting back the population and changing
the entire balance.
Hopefully, we will find solutions before life on earth is pushed to extinction.
Numericals
1. Starting from rest an automobile of mass 500 kg accelerates at 1.5 m/s2 for 12.0 second and
then continues to move at constant speed for 5 minutes. It is then stopped by a constant
retardation of 2 ms-2. Assuming that the whole journey is performed on a level road, the effect
of friction being equivalent to a constant retarding force of 100 N, calculate:
a. the total distance covered
b. the engine power used during motion at constant speed
c. during accelerated motion what is the
i.
resultant force acting on the automobile
ii.
average engine power used
[5589m;1800W;750N;7650W]
2. Refer to the graphs a] b] and c] given below, each of which shows variation of force F with
respect to displacement s in the direction of the force. For each case find the gain in kinetic
energy on covering a displacement of 4.0 metres.
3. The effective length of a simple pendulum is 1.0 m and the mass of its bob is 40g. The bob is
displaced through an angle of 60o and released. Give a sketch showing the bob in a displaced
position and mark in all forces acting on the bob. Which of these forces does work as the bob
is released from this displaced position? Find the kinetic energy when the string makes an
angle of 30o with the vertical.
[5.2 x 10-2 J]
4. A 12 kg block is pushed up through 20 m along sloping surface of a plane inclined at 37 o to the
horizontal by a constant force F of 120 N acting along the plane and parallel to the
displacement. If the frictional resistance is 24 N, find:
i.
ii.
iii.
iv.
v.
vi.
The work done by the applied force F
The work done by the frictional force f
The work done by the normal reaction
The work done by the weight of the body
The increase in the potential energy of the block
The increase in kinetic energy of the block.
[2400J; -480J; 0J; -1440J; +1440 J; +480 J]
5. The system shown below is released from rest when the 12 kg block is 3.0 m above the floor.
Use the principle of conservation of energy to find the velocity with which the block strikes the
floor. The string is light and there is no friction between the pulley and its axle.
[5.5 ms-1]
6. Explain with reason whether the law of conservation of energy is violated in each of the
following cases:a. Raindrops fall through air with constant velocity.
b. When a bullet is fired from a gun, both the bullet and the gun gain in kinetic energy
starting from rest.
c. When one runs up and down stairs one gets exhausted even though the net
mechanical work done is zero in each repeat.
d. When a helium filled balloon is released, it rises upwards and moves faster at the
sametime.