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COMPOSSED AND WRITTEN BY PROF. NAJEEB MUGHAL. GOVT. MUSLIM SCIENCE DEGREE COLLEGE HYD.
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CHAPTER 7
WORK, POWER AND ENERGY
Contents:
1. WORK AND UNITS.
2. GRAVITATIONAL FIELD IS CALLED COSERVATIVE FIELD.
3. POWER AND UNITS.
4. ENERGY AND KINDS OF ENERGY
5. KINETIC ENERGY.
6. POTENTAL ENERGY.
7. ABSOLUTEVPOTENTIAL ENERGY.
8. INTERCINVERSION OF ENERGY
9. LAW OF CONSERVATION OF ENERGY
10. EQUATIONS
11. DIMENSIONS
12. SHORT QUESTIONS AND ANSWERS.
DESCRIPTIVE
PART
1: Work:

The work done by an object with a constant force is the dot product of
the force F and the displacement d .
Work = F. d
Explanation:
Suppose an object undergoes a displacement d along a straight line while acted
on by a constant force F that makes an angle θ; its magnitude given by,
Work = Fd Cos 
This shows that work is a scalar quantity. The unit of work is “Joule” in S.I. or in M.K.S. and “erg” in C.G.S. system, have
dimension  M L2 T -2  .
Special cases:
Case .1
When,  = 0° Force and displacement are in the same direction.
Then,
Work = F d Cos 0°
Work = F d ( 1 )
Work = F d
This shows that, work is positive.
Case.2
When,  = 90°, Force and displacement are perpendicular to one another.
Then,
Work = F d Cos 90°
Work = F d (0)
Work = 0
this shows that, no work is to be done.
Case.3
When,  =180°,
Force and displacement are in opposite direction.
Then,
Work = F d Cos 180°
Work = F d (-1)
Work = - F d
this show that, work is negative.
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Joule: 

A constant force of one Newton acts parallel to a displacement of one meter; the work is
called “One Joule”.
Work = F d
1 Joule = 1N.1m
Erg: 

A constant force of one dyne acts parallel to a displacement of one centimeter; the work is
called “One Erg”.
Work = F d
1 Erg = 1dyne .1cm
Conversion between joule and Erg: 
We know that, Work = F d
Work = 1N.1m
1 N.1m = 1kg.1m sec-2 .1m
1 N.1m = 1000 gm.100 cm sec-2 (100 cm)
1 N.1m = 107 gm. cm sec-2 (cm)
1 N.1m = 10-7 dynes .cm
1 Joule = 10-7 Erg
1 joule = 10-7 erg= 0.7376 foot –pound
Electron volt : 

An electron volt is a measure of energy. An electron volt is the kinetic energy gained
by an electron passing through a potential difference of one volt “An electron volt (eV) is the
energy that an electron gains when it travels through a potential of one volt”.
Relation between electron volt (eV) and joule: 
We know that, V =
Work
q
Energy (Work) = Vq
1 electron volt (eV) = potential difference  charge of an electron.
1eV = V (1.6  10-19 C)
J
1eV= (1.610-19 C )
C
1eV = 1.6  10-19 J
2: Work done by a variable force (one dimensional case):
Let us consider a situation where the force is acting along the X-axis and the magnitude of
the force is varying with position 'x'. Thus, as the ball moves, the magnitude of the work done by
the force, on the ball, changes. The adjacent graph shows the plot of a one dimensional variable
force.
Let us calculate the work done on the ball by this force. We divide the area under the graph into
number of narrow strips of width 'Dx'. It is small enough to assume that F (x) is uniform in that
range. Let Fja be the average value of F (x) within the jth interval. Therefore, Dwj is the work done in the jth interval time; Dwj =
FjaDx Total work, W = Dwj = FjaDx
Work = lim  FΔx
Δx 0
Work =  F  x  dx
Geometrically, the work is equal to the area between the F(x) curve and the X-axis between the limits xi and xj.
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3:  Show that Gravitational field is also called Conservative field: 
Gravitational field:

A gravitational field is the force field created around massive bodies that causes
attraction of other massive bodies.
Conservative field:

A gravitational field of force in which the work done is moving a body from one
point to another is independent to the path taken. And other is the total amount of work
done in the closed path, in the gravitational field is equal to zero. Such gravitational field is
called, “Conservative Field”.
The force required to move the body between these points in a conservative field is called a
conservative force.
Proof of First part :
Work done is independent of path followed between two points:
Consider a body of weight W initially at rest is to be located at position P in the gravitational
field. The same is to be, moved through displacement d1 to position Q. The weight and
displacement are making an angle  with each other.
Work = F. d1
Therefore,
PQ
Work = W d1 Cos α
PQ
 Base 
Work = W d1 

P Q
 Hyp. 
d
Work = W d1  
P Q
 d1 
Work = W d
PQ
Again, the body is to be moved from position Q to S, in the gravitational field, through displacement d 2 at an angle 90°
made with the weight.
Work = W. d 2
Q S
Work = W d 2 Cos 90o
QS
Work = W d 2  0 
QS
Work = 0
QS
And, now again a body is to be moved from the initial position P to same final position S, in the gravitational field, through
displacement d3 at an angle  with the weight.
Work = W. d3
P S
Work =W d 3 Cos β
P S
 Base 
Work = W d3 

P S
 Hyp. 
d
Work = W d3  
P S
 d3 
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Work = W d
P S
Thus,
Work = Work + Work
P S
P Q
Q S
Wd=Wd+0
Or
W d = W d,
this shows that;

“The work done between two position points in the gravitational field is
independent of path followed the points”.
Proof of second Two:
Work done in closed path or loop is equal to zero:
Consider a body of weight W initially lying at rest acted vertically down ward, when placed at position P in the
gravitational field. It is moved toward Q through displacement d1 making an angle  with the weight of a body.
Work = F. d1
We know that,
PQ
Work = W d1 Cos α
PQ
 Base 
Work = W d1 

P Q
 Hyp. 
d
Work = W d1  
P Q
 d1 
Work = W d
PQ
Now a body is to be moved horizontally from position Q to S through displacement
d 2 at right angle to weight.
Work = W. d 2
Q S
Work = W d 2 Cos 90o
QS
Work = W d 2  0 
QS
Work = 0
QS
In the last a body is to be moved from position S to its final position P from which it starts its motion, upward in the gravitational
field, through displacement d3 making an angle (180° - ) with the weight.
Work = W. d3
S P
Work =W d3 Cos 180 - β 
S P
Work = W d3
SP
 Cos180° Cosβ
Work = W d3
SP
Work = W
SP
+ Sin 180o Sin β

 -1  Cos β + 0
d3  -Cos β 
Work = - W d 3 Cos β
SP
 base 
Work = - W d 3 

S P
 hyp. 
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 d 
Work = - W d3 

S P
 d3 
Work = - W d
S P
Total Work done= work + work + work
P Q S P
PQ
Total Work done=
P Q S P
QS
 W d  +  0  - W d 
Total Work done=
P Q S P
Total Work done = 0

SP
W d -  W d
This shows that,
“Total work done in a closed path or loop in the gravitational field is equal to zero”.
3: Power: 

The work done by a body, in unit time is called “Power”.
Work
Power =
time
Thus, power of a body is the “Rate of doing work”.
d 
F. d
Power =
or Power = F .  
t
t
Power = F . v
Power = F v Cos 
The power can be defined as,

“The dot product between force and velocity”. It is scalar quantity, have unit “watt”. Its
dimension is  M L2 T -3 
Watt: 

When, One-joule work is to be done by a body in one second then power is called
“One watt”.
work
Power =
time
1 joule
One watt =
1sec
1horse-power = 746 watts
And, 1horse power = 550 foot pound sec-1.
Convert one kilowatt-hour into joule: 
We know that,
Work = power (time)
1kilowatt hour =1000 Watt (3600 sec)
1 kilo watt hour = 36 105 (Joule / sec) sec
1kilo watt - hour = 3.6  106Joules
The kilowatt-hour is a Commercial unit of electrical energy.
5: Energy: 
A body which has the capacity to do work is said to possess energy. Whenever a body performs work it is by virtue of energy. No
energy no work. Energy is scalar quantity. There are number of kinds of sources of energy, for example:
Mechanical energy:
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Chemical energy:
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
The source of this type of energy is steam engines, which works on heat energy.

The source of this type of energy is our foods which makes grow and also gives
muscular energy to move .The other source is the burning chemical propellant.

The source of this type of energy is coil or sun or burning of chemical propellant.

The source of this type of energy is coil or electric sources.

The source of this type of energy is sound wave motion of living or non-living bodies.
Heat energy:
Light energy:
Sound energy:
Electrical energy:
 The source of this type of energy is electric motor.
Hydroelectric energy:
 The source of energy for the production of electricity by generators driven by water turbines.
Thermonuclear energy:
 The energy obtained by fusion f hydrogen nuclei to form heaver nucleus. It is called thermonuclear reaction.
Geo energy:
 The source of this type of energy is earth.
Solar energy:
 The source of this type of energy is mainly only sun.
Atomic energy:
The source of this type of energy is atoms.
Wind energy:
 The source of this type of energy is wind.
Tidal energy:

The source of this type of energy is water waves.
Fossil energy:

The source of this type of energy is old bones of animals
Nuclear energy:

energy released in nuclear reactions. When a neutron splits an atom's nucleus into smaller pieces it
is called fission. When two nuclei are joined together fewer than millions of degrees of heat it is called fusion
Electrical power.
 And, there are many other kinds of energy from them Mechanical energy is one. It is classified in two
parts; one of them is kinetic and other potential energies:
Kinetic energy:
 The source of energy is due to motion.
Potential energy:
 The source of energy is due to height or level.
All these sources of energy are used to perform work.
6: Calculation of kinetic energy: 
 The energy possessed due to motion is called “kinetic energy”.
Explanation:
Let us consider a body of mass is thrown up ward in the opposite direction of pull of gravity, in the gravitational field,
with initial velocity “v” to reach at height “h”. The work is to be done by the body is against the gravitational force.
Mathematical derivation:
Work = F . d
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Work = W h Cos 
Work = m g h Cos 0°
Work = m g h - - - - - - - - -- ►eq. (1)
If initial velocity of an object is”v”, its final velocity becomes zero at the height “h”.
Therefore, 2 g h =  vf2 - vi2 
2 (-g) h = 0 - v²
-2 g h = - v²
h=
1  v² 
2  g 
1  v² 
2  g 
1
Work =
 m v² 
2
The amount work done by virtue of motion of a body, is called “kinetic energy”.
Work = k. E.
1
K. E. =
 m v² 
2
By putting “h” in equation # 1, we get,
Work = m g
7: Calculation of Gravitational potential energy: 
 The energy possessed due to position is called “Gravitational potential energy”.
Explanation:
A body of mass “m” is at rest on the earth's surface. It is then raised a vertical distance “h” above the surface. The work
is to be done on to the body against the pull of gravity in the gravitational field. The energy, which stored in the body, is called
“Gravitational potential energy.” When external force acted which equal to the weight of a body then, works is done by the external
force is given by,
Mathematical derivation:
Work ext . = Fext . h
Workext. = m g h Cos 0°
Workext. = m g h
Work done by the gravitational force,
WorkG = - workext
The work done on to the body by an external force in the gravitation that stored in the form of “gravitational potential energy”
Thus,
UG = workext= - WorkG= - (- m g h )
Or
Gravitational potential energy = UG = m gh
8 : Absolute potential energy: 

The “Absolute potential energy” is the total energy, which the body possesses at any
point in the gravitational field.
Mathematical derivation:
For calculating the absolute potential energy at any point in the gravitational field, the displacement from the center of the earth
to such arbitrary point divided in to smaller equal displacement elements “r”. According to the Newton’s gravitational
m Me
force, F = G
on to body of mass m, due to Me, at a distance “r”.
r2
 r = r2 - r1
mM
The force, F1 = G 2 e , is exerted at a distance r1
r1
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m Me
, at a distance r2.
r22
The work due to average force between these two points is given by,
Work = Fave . r
Work = Fave r Cos 
Work = Fave r Cos 0°
because, Cos 0°=1
Work = Fave r ---------------► eq.# 1
F + F2
Fave = 1
2
G m Me
G m Me
+
2
r1
r22
Fave =
2
1 1
1
Fave = G m Me  2 + 2 
2
 r1 r2 
and, F2 = G
 r2 + r2 
1
G m Me  2 2 2 1 
2
 r1 r2 
  r + 1r 2 + r12 
1
= G m Me 

2
r12 r22


  r 2 +2r r1 + r12  + r12 
1

Fave = G m Me 
2
r12 r22


  0 +2  r2 - r1  r1 + r12  + r12 
1

Fave = G m Me 
2
r12 r22


  2r2 r1  2r12 + r12  + r12 
1

Fave = G m Me 
2
r12 r22


  2r2 r1  r12  + r12 
1

Fave = G m Me 
2
r12 r22


 2r r 
Fave = G m Me  22 12 
 2 r1 r2 
Fave =
Fave
 1 
Fave = G m Me 

 r1 r2 
The work done from the position 1 to position 2, is given by,
 1 
From equation # (1)
 G m Me 
 r
Work
12
 r1 r2 
Work 
12
 1 
G m Me 
  r2 - r1 
 r1 r2 
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 r
G m Me  2 12
 r2 r1
1
 G m Me  Work
12
 r1
Work 
r1 

r2 r1 
1

r2 
1 1
G m Me  - 
23
 r2 r3 
--- ----- - - - -  1
1
 G m Me 
- 
Work
N-1  N
 rN-1 rN 
Work Done  Work + work + - - - +
Work 
Similarly,
Total
1N
Total Work Done
1N
Total Work Done
1N
12
 1
 G m Me   r1
 1
 G m Me   r1
Total Work Done
1N
23
Work
N-1  N
 1
1 1 1
1 

 -  + - - - + 
r2   r2 r3 
 rN-1 rN  
 1
  1 1 
1 
- 
 + ---+ 
   r2 r3 
 rN-1 rN 
 



 1 
 1 
 G m Me   + - - - +  
 rN  
 r1 
1
r2
 1 
 1 
 G m Me   + - - - +  -  
  
1N
 r1 
1 1
 0
The Nth step in the gravitational field is at infinity .So that,
rN 
Total Work Done
1
 G m Me  
1N
 r1 
Thus, total work done in moving the body in the gravitational field is called” Absolute potential energy”.
Total work done = - (Absolute potential energy)
GmM e
Hence, A.P.E.= , is the potential energy on to earth surface.
Re
Because, r1 ≈ Re, is the radius of the earth.
Calculation of Absolute potential energy at (Re+h):
If, a body of mass “m” is to be moved at a height (Re+h), from the center of earth,
GmM e
GmMe
A.P.E.= Then, A.P.E.= R
Re  h 
Re  h e
Re
GmMe
A.P.E.=  Re h 


 Re
 Re Re 
Therefore,
Total Work Done
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A.P.E.= -
G m Me

h 
1 
 Re
 Re 
G m Me 
h 
A.P.E.= 1 

Re  Re 
A.P.E.= A.P.E.= -
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1

G m Me 
h
- - -  by binomial series
1 
Re  Re

G m Me 
h 
1 
 at any height “h” above the earth surface.
Re  Re 
: Inter- conversion of potential and kinetic energies: 
Explanation:
Suppose a body of mass “m” lifted vertically upward, the work done on to a body that stored in the form of gravitational
potential energy. A body in the state of rest at height “h” above the surface of ground say, position P will possesses,
Potential energy = mgh, And kinetic energy = 0
While falling a body, at position O, loss of potential energy and there is equal amount of gain of kinetic energy, due to
increase in its motion due to gravity.
P.E. Loss = K.E. Gain at the position O, as height ( h - x ).
1
W ( h - x ) = m v²
2
1
Wh-Wx =
m v²
2
1
m g h - m g x = m v²
2
1
m g h = m v² + m g x
2
Loss of P.E. = Gain of K.E. + Work done against force of friction
This equation is called “Work energy equation”.
1
If, f = 0 then, m g h = m v² + 0
2
1
m g h = m v²
2
P.E. Loss = K.E. Gain this is law of conservation of energy.
When force of friction is absent, the motion of body is independent of its mass.
: Law of conservation of energy: 

The principle of conservation of energy state that:”the total energy of a
system remains constant. Energy cannot be created or destroyed but may be converted
from one form to another”.
Total Work done (or Total Energy) = P.E. + K.E.
Explanation:
Sum of kinetic and potential energies of a free falling body remains constant. It means that kinetic energy is conserved just
before hitting the ground that is an equal amount of gravitational potential energy acquired by a body at the highest position.
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Let us consider a body of mass ”m” kept rest above the ground level at height “h”. The body
possesses potential energy at highest position P.
P.E.P = m g h and K.E.P = 0
Total energy P =P.E P + K.E P
Thus,
Total energy P = m g h + 0
T.EP= m g h
After dropping a body, at poison O, and body covers displacement “x”.
Thus,
P.E O = m g (h – x)
We know that, 2 g h = v f2 - vi2
2 g h = v² - 0
2 g h = v²
If, h = x, then, 2 g x = v²
1
K.E. = m v²
As,
2
1
Therefore K.E R = m ( 2 g x )
2
K.E R = ( m g x )
K.E.O = m g X, kinetic energy in terms of height x.
T.E.o = K.E.O + P.E.O
T.E.o= m g x + m g ( h - x )
T.E.o= m g x+ m g h - m g x
T.Eo = m g h
Finally, at position R, just before hitting the ground surface, K.E.R = m g h , is the kinetic energy in terms of height h.
P.E.R = 0
2 g h = vf2 - vi2
As,
2 g h = v2 - 0
2 g h = v²
1
1
m v ² = m 2 g h 
Where T.E. R =
2
2
T.E. R = K.E.R + P.E.R
T.E. R = m g h + 0
T.ER = m g h
Thus, total energy of a body is conserved during free falling at every position. This is “Law of conservation of energy.” There are
number of examples of law of conservation of energy in our daily life.
Equations
1. Work = F d Cos
7.Energy = Work
2. Work = F . d
8. K.E=
1
m v2
2
3. Power= F . v
4.Power= F v cos 5.Power=
9. P.E= mgh
12. Loss of P.E. = Gain of K.E. + Work done against force of friction
work
Time
10. Work = K.E + P.E.
13. A.P.E.= -
6.work=Power (time)
11. K.E. loss = P.E.gain
GmM e
Re
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Dimensions
PHYSICAL QUANTITY & SYMBOL
Work = F d
Power = F v
K.E.=P.E.= Work = ½ mv2 = m g h
DIMENSION
[ M L2 T-2]
2 -3
[ML T ]
[ M L2 T-2]
UNIT
Joule
Watt
Joule
Short questions
Q. # 1:
Answer:
We know that,
Convert one kilowatt hour in to joules?
Work = power (time)
Work = Kilowatt (hour)
Work = 1000 watt (3600sec)
Work = 3.6 x106Jsec-1 Sec
1Kilowatt.hour = 3.6x106Joules
Thus,
Q. # 2:
Answer:
Q. # 3:
Answer:
Define watt.
When one joule work is to be done in one second, then power is called one watt.
1Joule
Power =
1sec
1watt = 1joule.sec-1
Define joule
When one Newton force applied on a body and it covers one meter displacement in the direction of applied
force then work is called “One Joule”.
Work =1N. 1m
One joule = 1N.1m
Q. # 4:
Define power,
Answer:
When work is to be done in one second then it is called power.
Power = work / time
Q. # 5:
Define conservative field.
Answer:
Total work done in a closed path in the gravitational field is equal to zero and work done is independent between
two points in the gravitational field. Such gravitational field is called conservative field.
Q. # 6:
A block of wood is to be taken to the bottom of lake. Does the block possess potential energy?
Answer:
The block of wood possesses potential energy, when it is taken to the bottom of lake.
Q. # 7:
Does hydrogen filled balloon possess any potential energy?
Answer:
Yes, hydrogen filled balloon posse’s potential energy, because it possesses work due to height.
Q. # 8:
Define energy. Is it a scalar quantity?
Answer:
The ability of a body to do work is called “energy”. It is scalar quantity.
Q. # 9:
Write down the equation of potential energy for mass “m” lying on the surface of earth?
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Answer:
G m Me
Re
What will be power, if 2000J of work are done in 14 seconds?
Absolute potential energy = Q. # 10:
Answer:
Power =
Work
time
1200 J
14 sec
Power = 85.7 watt
Establish a relation between force and velocity.
Power=
Q. # 11:
Answer:
Work
time
F. d
Power =
t
Power = F . v , is the relation between force and velocity, in terms of power.
Q. # 12:
What do you know about i) positive work ii) zero work iii) negative work?
Answer:
i)
Work will be positive, when the force and displacement are in the same direction.
i)Work = F d cos θ
Work = F d Cos 0o
Work = F d
ii)
Work will be zero, when the force and displacement are perpendicular on each other.
Work = F d cos θ
Work = F d Cos 90o
Work = F d (0)
Work = 0
iii)
Work will be Negative, when the force and displacement are in the opposite direction.
Work = F d cos θ
Work = F d Cos 180o
Work = F d ( - 1)
Work = - F d
Q. # 13:
What happens to the kinetic energy of a bullet, when it penetrates in to target?
Answer:
When a bullet penetrates in to target then kinetic energy performs work wile penetrating against opposing
forces. Thus, K.E = Work
Q. # 14:
What form of energy is produced, when work is to be done against friction?
Answer:
When work is to be done against friction Heat energy produced. Thus, Work = Heat energy
Q. # 15:
What type of energy is to be stored in the spring of watch?
Answer:
The energy stored in the stretched spring of watch is potential energy.
Q. # 16:
Define potential energy, kinetic energy
Power =
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Answer:
The energy possesses by a body due to position is called “Potential energy”. And
energy possesses by a body due to motion is called “Kinetic energy”.
Q. # 17:
You are sitting in a moving car .Do you possess any potential or kinetic energy with
reference to
i) the car ii) the earth?
Answer:
As I am sitting in a moving car, I possess potential energy with respect to position of
earth and, with respect to surrounding of moving car kinetic energy.
Q. # 18: As a pendulum bob swings, at what point is its kinetic energy maximum? At what point is its potential
energy maximum?
Answer:
As a pendulum bob swings, it possesses kinetic energy maximum, at the mean position. And potential energy
maximum possesses, at the extreme potion either at right or left of mean position.
Q. # 19: Why do the bob of swinging pendulum come to rest after some time? What does happen to its energy?
Answer:
The bob of swinging pendulum come to rest after some time due to different frictions. Its energy is
converted to perform work against different frictions and heat energy.
Q. # 20: Does the tension in the string of swinging pendulum do any work?
Answer:
No, the tension in the string of swinging pendulum doesn’t do any work.
Q. # 21: Is kinetic energy vector quantity?
Answer:
No, kinetic energy is not vector quantity.
Q. #22: A 2.0 kg block is thrown upward from a point 20m above Earth's surface. At what height above Earth's
surface will the gravitational potential energy of the Earth block system have increased by 500J?
Answer:
Mass of the block m = 2.0 kg
P.E. of the block at a height of 20 m is = mgh = 2 x 9,8 x 20 = 392 J
Increase in P.E. = 500 J
Let the new height be h
New P.E. = mgh
mgh -392 = 500
mgh = 500 + 392 = 892 J
therefore h = 892/2x9.8 = 45.51m
Q. # 23: what point the potential energy of a body is taken to be zero?
Answer:
The potential energy of the body at the surface of the earth is taken to be zero (Potential energy = mgh,
where h is the height of the body from the surface.).
Q. # 24: Does the work done on a body by a force depend upon the path followed by it?
Answer:
May or may not be. If the force is conservative then it does not depend but if it is non-conservative (friction) then it
depends.
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