# work, energy and power

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```CHAPTER SIX
WORK, ENERGY
AND
POWER
6.1 INTRODUCTION
6.1 Introduction
6.2 Notions of work and kinetic
energy : The work-energy
theorem
6.3 Work
6.4 Kinetic energy
6.5 Work done by a variable
force
6.6 The work-energy theorem for
a variable force
6.7 The concept of potential
energy
6.8 The conservation of
mechanical energy
6.9 The potential energy of a
spring
6.10 Various forms of energy : the
law of conservation of energy
6.11 Power
6.12 Collisions
Summary
Points to ponder
Exercises
Appendix 6.1
The terms ‘work’, ‘energy’ and ‘power’ are frequently used
in everyday language. A farmer ploughing the field, a
construction worker carrying bricks, a student studying for
a competitive examination, an artist painting a beautiful
landscape, all are said to be working. In physics, however,
the word ‘Work’ covers a definite and precise meaning.
Somebody who has the capacity to work for 14-16 hours a
day is said to have a large stamina or energy. We admire a
long distance runner for her stamina or energy. Energy is
thus our capacity to do work. In Physics too, the term ‘energy’
is related to work in this sense, but as said above the term
‘work’ itself is defined much more precisely. The word ‘power’
is used in everyday life with different shades of meaning. In
karate or boxing we talk of ‘powerful’ punches. These are
delivered at a great speed. This shade of meaning is close to
the meaning of the word ‘power’ used in physics. We shall
find that there is at best a loose correlation between the
physical definitions and the physiological pictures these
terms generate in our minds. The aim of this chapter is to
develop an understanding of these three physical quantities.
Before we proceed to this task, we need to develop a
mathematical prerequisite, namely the scalar product of two
vectors.
6.1.1 The Scalar Product
We have learnt about vectors and their use in Chapter 4.
Physical quantities like displacement, velocity, acceleration,
force etc. are vectors. We have also learnt how vectors are
added or subtracted. We now need to know how vectors are
multiplied. There are two ways of multiplying vectors which
we shall come across : one way known as the scalar product
gives a scalar from two vectors and the other known as the
vector product produces a new vector from two vectors. We
shall look at the vector product in Chapter 7. Here we take
up the scalar product of two vectors. The scalar product or
dot product of any two vectors A and B, denoted as A.B (read
WORK, ENERGY AND POWER
A dot B) is defined as
A.B = A B cos θ
115
˜
A = Ax ˜i + A y ˜j + Az k
(6.1a)
where θ is the angle between the two vectors as
shown in Fig. 6.1(a). Since A, B and cos θ are
scalars, the dot product of A and B is a scalar
quantity. Each vector, A and B, has a direction
but their scalar product does not have a
direction.
From Eq. (6.1a), we have
A .B = A (B cos θ )
= B (A cos θ )
Geometrically, B cos θ is the projection of B onto
A in Fig.6.1 (b) and A cos θ is the projection of A
onto B in Fig. 6.1 (c). So, A.B is the product of
the magnitude of A and the component of B along
A. Alternatively, it is the product of the
magnitude of B and the component of A along B.
Equation (6.1a) shows that the scalar product
follows the commutative law :
A.B = B.A
Scalar product obeys the distributive
law:
A . (B + C) = A .B + A .C
Further, λ A. (λ B) = λ (A.B)
˜
B = B x ˜i + B y ˜j + B z k
their scalar product is
(
)(
˜ . B ˜i + B ˜j + B k
˜
A.B = A x ˜i + Ay ˜j + Az k
x
y
z
= A x B x + Ay By + Az B z
(6.1b)
From the definition of scalar product and
(Eq. 6.1b) we have :
(i)
A . A = A x A x + Ay Ay + A z A z
2
\$j ⋅ \$j = k\$ ⋅ k\$ = 1
\$j ⋅ k\$ = k\$ ⋅ \$i = 0
Given two vectors
2
2
u Example 6.1 Find the angle between force
ˆ unit and displacement
F = (3 \$i + 4 \$j – 5 k)
ˆ unit. Also find the
d = (5 \$i + 4 \$j + 3 k)
projection of F on d.
Answer F.d = Fx d x + Fy d y + Fz d z
= 3 (5) + 4 (4) + (– 5) (3)
= 16 unit
Hence F.d
= F d cos θ = 16 unit
Now F.F
= F 2 = Fx2 + Fy2 + Fz2
= 9 + 16 + 25
= 50 unit
and d.d
= d 2 = d x2 + dy2 + dz2
= 25 + 16 + 9
= 50 unit
For unit vectors \$i, \$j, k\$ we have
i\$ ⋅ \$i =
i\$ ⋅ \$j =
2
A = A x + Ay + A z
(6.1c)
since A .A = |A ||A| cos 0 = A2.
(ii)
A .B = 0, if A and B are perpendicular.
Or,
where λ is a real number.
The proofs of the above equations are left to
you as an exercise.
)
∴ cos θ
=
16
50 50
=
16
= 0.32 ,
50
θ = cos–1 0.32
Fig. 6.1 (a) The scalar product of two vectors A and B is a scalar : A. B = A B cos θ. (b) B cos θ is the projection
of B onto A. (c) A cos θ is the projection of A onto B.
116
PHYSICS
6.2 NOTIONS OF WORK AND KINETIC
ENERGY: THE WORK-ENERGY THEOREM
The following relation for rectilinear motion under
constant acceleration a has been encountered
in Chapter 3,
v2 − u2 = 2 as
where u and v are the initial and final speeds
and s the distance traversed. Multiplying both
sides by m/2, we have
1
1
mv 2 − mu 2 = mas = Fs
2
2
known to be proportional to the speed of
the drop but is otherwise undetermined.
Consider a drop of mass 1.00 g falling from
a height 1.00 km. It hits the ground with
a speed of 50.0 m s-1. (a) What is the work
done by the gravitational force ? What is
the work done by the unknown resistive
force?
Answer (a) The change in kinetic energy of the
drop is
(6.2a)
∆K =
where the last step follows from Newton’s
Second Law. We can generalise Eq. (6.1)
to three dimensions by employing
vectors
v2 − u2 = 2 a.d
Once again multiplying both sides by m/2 , we
obtain
1
1
mv 2 − mu 2 = m a.d = F.d
(6.2b)
2
2
The above equation provides a motivation for
the definitions of work and kinetic energy. The
left side of the equation is the difference in the
quantity ‘half the mass times the square of the
speed’ from its initial value to its final value. We
call each of these quantities the ‘kinetic energy’,
denoted by K. The right side is a product of the
displacement and the component of the force
along the displacement. This quantity is called
‘work’ and is denoted by W. Eq. (6.2b) is then
Kf − Ki = W
1
× 10-3 × 50 × 50
2
= 1.25 J
=
where we have assumed that the drop is initially
at rest.
Assuming that g is a constant with a value
10 m/s2, the work done by the gravitational force
is,
Wg = mgh
= 10-3 ×10 ×103
= 10.0 J
(b) From the work-energy theorem
∆ K = W g + Wr
where Wr is the work done by the resistive force
on the raindrop. Thus
Wr = ∆K − Wg
= 1.25 −10
= − 8.75 J
(6.3)
is negative.
where Ki and Kf are respectively the initial and
final kinetic energies of the object. Work refers
to the force and the displacement over which it
acts. Work is done by a force on the body over
a certain displacement.
Equation (6.2) is also a special case of the
work-energy (WE) theorem : The change in
kinetic energy of a particle is equal to the
work done on it by the net force. We shall
generalise the above derivation to a varying force
in a later section.
6.3 WORK
⊳ Example 6.2 It is well known that a
raindrop falls under the influence of the
downward gravitational force and the
opposing resistive force. The latter is
1
m v2 − 0
2
⊳
As seen earlier, work is related to force and the
displacement over which it acts. Consider a
constant force F acting on an object of mass m.
The object undergoes a displacement d in the
positive x-direction as shown in Fig. 6.2.
Fig. 6.2 An object undergoes a displacement d
under the influence of the force F.
WORK, ENERGY AND POWER
117
The work done by the force is defined to be
the product of component of the force in the
direction of the displacement and the
magnitude of this displacement. Thus
(6.4)
We see that if there is no displacement, there
is no work done even if the force is large. Thus,
when you push hard against a rigid brick wall,
the force you exert on the wall does no work. Yet
your muscles are alternatively contracting and
relaxing and internal energy is being used up
and you do get tired. Thus, the meaning of work
in physics is different from its usage in everyday
language.
No work is done if :
(i) the displacement is zero as seen in the
example above. A weightlifter holding a 150
kg mass steadily on his shoulder for 30 s
does no work on the load during this time.
(ii) the force is zero. A block moving on a smooth
horizontal table is not acted upon by a
horizontal force (since there is no friction), but
may undergo a large displacement.
(iii) the force and displacement are mutually
perpendicular. This is so since, for θ = π/2 rad
(= 90o), cos (π/2) = 0. For the block moving on
a smooth horizontal table, the gravitational
force mg does no work since it acts at right
angles to the displacement. If we assume that
the moon’s orbits around the earth is
perfectly circular then the earth’s
gravitational force does no work. The moon’s
instantaneous displacement is tangential
while the earth’s force is radially inwards and
θ = π/2.
Work can be both positive and negative. If θ is
between 0o and 90o, cos θ in Eq. (6.4) is positive.
If θ is between 90o and 180o, cos θ is negative.
In many examples the frictional force opposes
displacement and θ = 180o. Then the work done
by friction is negative (cos 180o = –1).
From Eq. (6.4) it is clear that work and energy
have the same dimensions, [ML2T–2]. The SI unit
of these is joule (J), named after the famous British
physicist James Prescott Joule (1811-1869). Since
work and energy are so widely used as physical
concepts, alternative units abound and some of
these are listed in Table 6.1.
⊳
W = (F cos θ )d = F.d
Table 6.1 Alternative Units of Work/Energy in J
Example 6.3 A cyclist comes to a skidding
stop in 10 m. During this process, the force
on the cycle due to the road is 200 N and
is directly opposed to the motion. (a) How
much work does the road do on the cycle ?
(b) How much work does the cycle do on
the work done by the stopping (frictional) force
on the cycle due to the road.
(a) The stopping force and the displacement make
an angle of 180o (π rad) with each other.
Thus, work done by the road,
Wr = Fd cosθ
= 200 × 10 × cos π
= – 2000 J
It is this negative work that brings the cycle
to a halt in accordance with WE theorem.
(b) From Newton’s Third Law an equal and
opposite force acts on the road due to the
cycle. Its magnitude is 200 N. However, the
work done by cycle on the road is zero.
⊳
The lesson of Example 6.3 is that though the
force on a body A exerted by the body B is always
equal and opposite to that on B by A (Newton’s
Third Law); the work done on A by B is not
necessarily equal and opposite to the work done
on B by A.
6.4 KINETIC ENERGY
As noted earlier, if an object of mass m has
velocity v, its kinetic energy K is
K =
1
1
m v. v = mv 2
2
2
(6.5)
Kinetic energy is a scalar quantity. The kinetic
energy of an object is a measure of the work an
118
PHYSICS
Table 6.2 Typical kinetic energies (K)
object can do by the virtue of its motion. This
notion has been intuitively known for a long time.
The kinetic energy of a fast flowing stream
has been used to grind corn. Sailing
ships employ the kinetic energy of the wind. Table
6.2 lists the kinetic energies for various
objects.
Example 6.4 In a ballistics demonstration
a police officer fires a bullet of mass 50.0 g
with speed 200 m s-1 (see Table 6.2) on soft
plywood of thickness 2.00 cm. The bullet
emerges with only 10% of its initial kinetic
energy. What is the emergent speed of the
bullet ?
This is illustrated in Fig. 6.3(a). Adding
successive rectangular areas in Fig. 6.3(a) we
get the total work done as
xf
W≅
∑F (x )∆x
where the summation is from the initial position
xi to the final position xf.
If the displacements are allowed to approach
zero, then the number of terms in the sum
increases without limit, but the sum approaches
a definite value equal to the area under the curve
in Fig. 6.3(b). Then the work done is
⊳
W =
Answer The initial kinetic energy of the bullet
is mv2/2 = 1000 J. It has a final kinetic energy
of 0.1×1000 = 100 J. If vf is the emergent speed
of the bullet,
1
2
2
mv f
vf =
(6.6)
xi
lim xf
lim
F (x )∆x
∆x → 0
∑
xi
xf
=
∫ F ( x ) dx
(6.7)
xi
= 100 J
2 × 100 J
0.05 kg
= 63.2 m s–1
where ‘lim’ stands for the limit of the sum when
∆x tends to zero. Thus, for a varying force
the work done can be expressed as a definite
Appendix 3.1).
The speed is reduced by approximately 68%
(not 90%).
⊳
6.5 WORK DONE BY A VARIABLE FORCE
A constant force is rare. It is the variable force,
which is more commonly encountered. Fig. 6.2
is a plot of a varying force in one dimension.
If the displacement ∆x is small, we can take
the force F (x) as approximately constant and
the work done is then
∆W =F (x) ∆x
Fig. 6.3(a)
WORK, ENERGY AND POWER
119
The work done by the frictional force is
Wf → area of the rectangle AGHI
Wf = (−50) × 20
= − 1000 J
The area on the negative side of the force axis
has a negative sign.
⊳
6.6 THE WORK-ENERGY THEOREM FOR A
VARIABLE FORCE
Fig. 6.3 (a) The shaded rectangle represents the
work done by the varying force F(x), over
the small displacement ∆x, ∆W = F(x) ∆x.
(b) adding the areas of all the rectangles we
find that for ∆x → 0, the area under the curve
is exactly equal to the work done by F(x).
Example 6.5 A woman pushes a trunk on
a railway platform which has a rough
surface. She applies a force of 100 N over a
distance of 10 m. Thereafter, she gets
progressively tired and her applied force
reduces linearly with distance to 50 N. The
total distance through which the trunk has
been moved is 20 m. Plot the force applied
by the woman and the frictional force, which
is 50 N versus displacement. Calculate the
work done by the two forces over 20 m.
We are now familiar with the concepts of work
and kinetic energy to prove the work-energy
theorem for a variable force. We confine
ourselves to one dimension. The time rate of
change of kinetic energy is
dK
d 1
2
=
 m v 
dt
dt  2
dv
v
dt
=F v (from Newton’s Second Law)
=m
=F
dx
dt
Thus
dK = Fdx
Integrating from the initial position (x i ) to final
position ( x f ), we have
⊳
Kf
∫
Ki
xf
dK =
∫
Fdx
xi
where, Ki and K f are the initial and final kinetic
energies corresponding to x i and x f.
xf
Fig. 6.4 Plot of the force F applied by the woman and
the opposing frictional force f versus
displacement.
The plot of the applied force is shown in Fig.
6.4. At x = 20 m, F = 50 N (≠ 0). We are given
that the frictional force f is |f|= 50 N. It opposes
motion and acts in a direction opposite to F. It
is therefore, shown on the negative side of the
force axis.
The work done by the woman is
WF → area of the rectangle ABCD + area of
the trapezium CEID
1
WF = 100 × 10 + (100 + 50) × 10
2
= 1000 + 750
= 1750 J
or
K f − Ki =
∫ Fdx
(6.8a)
xi
From Eq. (6.7), it follows that
Kf − Ki = W
(6.8b)
Thus, the WE theorem is proved for a variable
force.
While the WE theorem is useful in a variety of
problems, it does not, in general, incorporate the
complete dynamical information of Newton’s
second law. It is an integral form of Newton’s
second law. Newton’s second law is a relation
between acceleration and force at any instant of
time. Work-energy theorem involves an integral
over an interval of time. In this sense, the temporal
(time) information contained in the statement of
Newton’s second law is ‘integrated over’ and is
120
PHYSICS
not available explicitly. Another observation is that
Newton’s second law for two or three dimensions
is in vector form whereas the work-energy
theorem is in scalar form. In the scalar form,
information with respect to directions contained
in Newton’s second law is not present.
Example 6.6 A block of mass m = 1 kg,
moving on a horizontal surface with speed
vi = 2 ms–1 enters a rough patch ranging
from x = 0.10 m to x = 2.01 m. The retarding
force Fr on the block in this range is inversely
proportional to x over this range,
⊳
−k
for 0.1 < x < 2.01 m
x
= 0 for x < 0.1m and x > 2.01 m
where k = 0.5 J. What is the final kinetic
energy and speed vf of the block as it
crosses this patch ?
Fr =
2.01
K f = Ki +
∫
0.1
=
( −k ) dx
x
1
mvi2 − k ln ( x )
2
2.01
0.1
1
= mvi2 − k ln (2.01/0.1)
2
= 2 − 0.5 ln (20.1)
= 2 − 1.5 = 0.5 J
v f = 2K f /m = 1 m s−1
Here, note that ln is a symbol for the natural
logarithm to the base e and not the logarithm to
the base 10 [ln X = loge X = 2.303 log10 X].
⊳
6.7 THE CONCEPT OF POTENTIAL ENERGY
The word potential suggests possibility or
capacity for action. The term potential energy
brings to one’s mind ‘stored’ energy. A stretched
bow-string possesses potential energy. When it
is released, the arrow flies off at a great speed.
The earth’s crust is not uniform, but has
discontinuities and dislocations that are called
fault lines. These fault lines in the earth’s crust
*
are like ‘compressed springs’. They possess a
large amount of potential energy. An earthquake
results when these fault lines readjust. Thus,
potential energy is the ‘stored energy’ by virtue
of the position or configuration of a body. The
body left to itself releases this stored energy in
the form of kinetic energy. Let us make our notion
of potential energy more concrete.
The gravitational force on a ball of mass m is
mg . g may be treated as a constant near the earth
surface. By ‘near’ we imply that the height h of
the ball above the earth’s surface is very small
compared to the earth’s radius RE (h <<RE) so that
we can ignore the variation of g near the earth’s
surface*. In what follows we have taken the
upward direction to be positive. Let us raise the
ball up to a height h. The work done by the external
agency against the gravitational force is mgh. This
work gets stored as potential energy.
Gravitational potential energy of an object, as a
function of the height h, is denoted by V(h) and it
is the negative of work done by the gravitational
force in raising the object to that height.
V (h) = mgh
If h is taken as a variable, it is easily seen that
the gravitational force F equals the negative of
the derivative of V(h) with respect to h. Thus,
d
V(h) = −m g
dh
gravitational force is downward. When released,
the ball comes down with an increasing speed.
Just before it hits the ground, its speed is given
by the kinematic relation,
v2 = 2gh
This equation can be written as
F =−
1
m v2 = m g h
2
which shows that the gravitational potential
energy of the object at height h, when the object
is released, manifests itself as kinetic energy of
the object on reaching the ground.
Physically, the notion of potential energy is
applicable only to the class of forces where work
done against the force gets ‘stored up’ as energy.
When external constraints are removed, it
manifests itself as kinetic energy. Mathematically,
(for simplicity, in one dimension) the potential
The variation of g with height is discussed in Chapter 8 on Gravitation.
WORK, ENERGY AND POWER
energy V(x) is defined if the force F(x) can be
written as
F (x ) = −
dV
dx
This implies that
xf
∫
xi
Vf
F(x) dx = − ∫ dV = Vi − V f
Vi
The work done by a conservative force such as
gravity depends on the initial and final positions
only. In the previous chapter we have worked
on examples dealing with inclined planes. If an
object of mass m is released from rest, from the
top of a smooth (frictionless) inclined plane of
height h, its speed at the bottom
is 2gh irrespective of the angle of inclination.
Thus, at the bottom of the inclined plane it
acquires a kinetic energy, mgh. If the work done
or the kinetic energy did depend on other factors
such as the velocity or the particular path taken
by the object, the force would be called nonconservative.
The dimensions of potential energy are
[ML2T –2] and the unit is joule (J), the same as
kinetic energy or work. To reiterate, the change
in potential energy, for a conservative force,
∆V is equal to the negative of the work done by
the force
∆V = − F(x) ∆x
(6.9)
In the example of the falling ball considered in
this section we saw how potential energy was
converted to kinetic energy. This hints at an
important principle of conservation in mechanics,
which we now proceed to examine.
6.8 THE CONSERVATION OF MECHANICAL
ENERGY
For simplicity we demonstrate this important
principle for one-dimensional motion. Suppose
that a body undergoes displacement ∆x under
the action of a conservative force F. Then from
the WE theorem we have,
∆K = F(x) ∆x
If the force is conservative, the potential energy
function V(x) can be defined such that
− ∆V = F(x) ∆x
The above equations imply that
∆K + ∆V = 0
∆(K + V ) = 0
(6.10)
121
which means that K + V, the sum of the kinetic
and potential energies of the body is a constant.
Over the whole path, xi to xf, this means that
Ki + V(xi ) = Kf + V(xf )
(6.11)
The quantity K +V(x), is called the total
mechanical energy of the system. Individually
the kinetic energy K and the potential energy
V(x) may vary from point to point, but the sum
is a constant. The aptness of the term
‘conservative force’ is now clear.
Let us consider some of the definitions of a
conservative force.
l
A force F(x) is conservative if it can be derived
from a scalar quantity V(x) by the relation
given by Eq. (6.9). The three-dimensional
generalisation requires the use of a vector
derivative, which is outside the scope of this
book.
l
The work done by the conservative force
depends only on the end points. This can be
seen from the relation,
W = Kf – Ki = V (xi ) – V(xf )
which depends on the end points.
l
A third definition states that the work done
by this force in a closed path is zero. This is
once again apparent from Eq. (6.11) since
xi = xf .
Thus, the principle of conservation of total
mechanical energy can be stated as
The total mechanical energy of a system is
conserved if the forces, doing work on it, are
conservative.
The above discussion can be made more
concrete by considering the example of the
gravitational force once again and that of the
spring force in the next section. Fig. 6.5 depicts
a ball of mass m being dropped from a cliff of
height H.
Fig. 6.5
The conversion of potential energy to kinetic
energy for a ball of mass m dropped from a
height H.
122
PHYSICS
The total mechanical energies E0, Eh, and EH
of the ball at the indicated heights zero (ground
level), h and H, are
(6.11 a)
EH = mgH
1
mvh2
2
= (1/2) mvf2
Eh = mgh +
(6.11 b)
E0
(6.11 c)
The constant force is a special case of a spatially
dependent force F(x). Hence, the mechanical
energy is conserved. Thus
EH = E0
1
mgH = mv 2f
or,
2
Answer (i) There are two external forces on
the bob : gravity and the tension (T ) in the
string. The latter does no work since the
displacement of the bob is always normal to the
string. The potential energy of the bob is thus
associated with the gravitational force only. The
total mechanical energy E of the system is
conserved. We take the potential energy of the
system to be zero at the lowest point A. Thus,
at A :
E=
v f = 2gH
1
2
TA − mg =
a result that was obtained in section 3.7 for a
freely falling body.
Further,
EH = Eh
which implies,
v 2h = 2 g(H − h)
Example 6.7 A bob of mass m is suspended
by a light string of length L . It is imparted a
horizontal velocity vo at the lowest point A
such that it completes a semi-circular
trajectory in the vertical plane with the string
becoming slack only on reaching the topmost
point, C. This is shown in Fig. 6.6. Obtain an
expression for (i) vo; (ii) the speeds at points
B and C; (iii) the ratio of the kinetic energies
(KB/KC) at B and C. Comment on the nature
of the trajectory of the bob after it reaches
the point C.
mv02
L
(6.12)
[Newton’s Second Law]
where TA is the tension in the string at A. At the
highest point C, the string slackens, as the
tension in the string (TC ) becomes zero.
Thus, at C
(6.11 d)
and is a familiar result from kinematics.
At the height H, the energy is purely potential.
It is partially converted to kinetic at height h and
is fully kinetic at ground level. This illustrates
the conservation of mechanical energy.
2
mv0
E=
1
mvc2 + 2mgL
2
mg =
mvc2
L
[Newton’s Second Law]
(6.13)
(6.14)
where vC is the speed at C. From Eqs. (6.13) and
(6.14)
5
mgL
2
Equating this to the energy at A
E =
5
m 2
mgL =
v0
2
2
or,
v0 = 5 gL
(ii) It is clear from Eq. (6.14)
vC = gL
At B, the energy is
E=
1
mv 2B + mgL
2
Equating this to the energy at A and employing
the result from (i), namely v 02 = 5 gL ,
1
1
mv 2B + mgL = mv02
2
2
Fig. 6.6
=
5
m gL
2
⊳
WORK, ENERGY AND POWER
123
∴ v B = 3gL
W =+
2
k xm
2
(6.16)
(iii) The ratio of the kinetic energies at B and C
is :
1 2
mv
KB 2 B 3
=
=
1 2 1
KC
mvC
2
At point C, the string becomes slack and the
velocity of the bob is horizontal and to the left. If
the connecting string is cut at this instant, the
bob will execute a projectile motion with
horizontal projection akin to a rock kicked
horizontally from the edge of a cliff. Otherwise
the bob will continue on its circular path and
complete the revolution.
⊳
6.9 THE POTENTIAL ENERGY OF A SPRING
The spring force is an example of a variable force
which is conservative. Fig. 6.7 shows a block
attached to a spring and resting on a smooth
horizontal surface. The other end of the spring
is attached to a rigid wall. The spring is light
and may be treated as massless. In an ideal
spring, the spring force Fs is proportional to
x where x is the displacement of the block from
the equilibrium position. The displacement could
be either positive [Fig. 6.7(b)] or negative
[Fig. 6.7(c)]. This force law for the spring is called
Hooke’s law and is mathematically stated as
Fs = − kx
The constant k is called the spring constant. Its
unit is N m-1. The spring is said to be stiff if k is
large and soft if k is small.
Suppose that we pull the block outwards as in
Fig. 6.7(b). If the extension is xm, the work done by
the spring force is
xm
xm
Ws =
∫
0
=−
2
k xm
2
Fs d x
=−
∫ kx dx
0
(6.15)
This expression may also be obtained by
considering the area of the triangle as in
Fig. 6.7(d). Note that the work done by the
external pulling force F is positive since it
overcomes the spring force.
Fig. 6.7
Illustration of the spring force with a block
attached to the free end of the spring.
(a) The spring force Fs is zero when the
displacement x from the equilibrium position
is zero. (b) For the stretched spring x > 0
and Fs < 0 (c) For the compressed spring
x < 0 and Fs > 0.(d) The plot of Fs versus x.
The area of the shaded triangle represents
the work done by the spring force. Due to the
opposing signs of Fs and x, this work done is
2
negative, Ws = −kx m
/ 2.
The same is true when the spring is
compressed with a displacement xc (< 0). The
spring force does work Ws = − kx c2 / 2 while the
124
PHYSICS
external force F does work + kx c2 / 2 . If the block
is moved from an initial displacement xi to a
final displacement xf , the work done by the
spring force Ws is
xf
Ws = − ∫ k x dx
=
xi
k x i2
2
−
k x 2f
and vice versa, however, the total mechanical
energy remains constant. This is graphically
depicted in Fig. 6.8.
(6.17)
2
Thus the work done by the spring force depends
only on the end points. Specifically, if the block
=
xi
k x i2
2
−
k x i2
2
=0
(6.18)
The work done by the spring force in a cyclic
process is zero. We have explicitly demonstrated
that the spring force (i) is position dependent
only as first stated by Hooke, (Fs = − kx); (ii)
does work which only depends on the initial and
final positions, e.g. Eq. (6.17). Thus, the spring
force is a conservative force.
We define the potential energy V(x) of the spring
to be zero when block and spring system is in the
equilibrium position. For an extension (or
compression) x the above analysis suggests that
kx 2
(6.19)
2
You may easily verify that − dV/dx = − k x, the
spring force. If the block of mass m in Fig. 6.7 is
extended to xm and released from rest, then its
total mechanical energy at any arbitrary point x,
where x lies between – xm and + xm, will be given by
V(x) =
1
1
2 1
k xm
= k x 2 + m v2
2
2
2
where we have invoked the conservation of
mechanical energy. This suggests that the speed
and the kinetic energy will be maximum at the
equilibrium position, x = 0, i.e.,
1
1
2
2
m vm
= k xm
2
2
where vm is the maximum speed.
or
vm =
k
xm
m
Note that k/m has the dimensions of [T-2] and
our equation is dimensionally correct. The
kinetic energy gets converted to potential energy
Fig. 6.8 Parabolic plots of the potential energy V and
kinetic energy K of a block attached to a
spring obeying Hooke’s law. The two plots
are complementary, one decreasing as the
other increases. The total mechanical
energy E = K + V remains constant.
⊳
xi
Ws = − ∫ k x dx
Example 6.8 To simulate car accidents, auto
manufacturers study the collisions of moving
cars with mounted springs of different spring
constants. Consider a typical simulation with
a car of mass 1000 kg moving with a speed
18.0 km/h on a smooth road and colliding
with a horizontally mounted spring of spring
constant 6.25 × 103 N m–1. What is the
maximum compression of the spring ?
Answer At maximum compression the kinetic
energy of the car is converted entirely into the
potential energy of the spring.
The kinetic energy of the moving car is
K =
=
1
mv 2
2
1
× 103 × 5 × 5
2
K = 1.25 × 104 J
where we have converted 18 km h–1 to 5 m s–1 [It is
useful to remember that 36 km h–1 = 10 m s–1].
At maximum compression xm, the potential
energy V of the spring is equal to the kinetic
energy K of the moving car from the principle of
conservation of mechanical energy.
1
2
V = k xm
2
WORK, ENERGY AND POWER
125
= 1.25 × 104 J
We obtain
xm = 2.00 m
We note that we have idealised the situation.
The spring is considered to be massless. The
surface has been considered to possess
negligible friction.
⊳
We conclude this section by making a few
remarks on conservative forces.
(i) Information on time is absent from the above
discussions. In the example considered
above, we can calculate the compression, but
not the time over which the compression
occurs. A solution of Newton’s Second Law
for this system is required for temporal
information.
(ii) Not all forces are conservative. Friction, for
example, is a non-conservative force. The
principle of conservation of energy will have
to be modified in this case. This is illustrated
in Example 6.9.
(iii) The zero of the potential energy is arbitrary.
It is set according to convenience. For the
spring force we took V(x) = 0, at x = 0, i.e. the
energy. For the constant gravitational force
mg, we took V = 0 on the earth’s surface. In
a later chapter we shall see that for the force
due to the universal law of gravitation, the
zero is best defined at an infinite distance
from the gravitational source. However, once
the zero of the potential energy is fixed in a
given discussion, it must be consistently
adhered to throughout the discussion. You
cannot change horses in midstream !
Example 6.9 Consider Example 6.8 taking
the coefficient of friction, µ, to be 0.5 and
calculate the maximum compression of the
spring.
⊳
Answer In presence of friction, both the spring
force and the frictional force act so as to oppose
the compression of the spring as shown in
Fig. 6.9.
We invoke the work-energy theorem, rather
than the conservation of mechanical energy.
The change in kinetic energy is
Fig. 6.9 The forces acting on the car.
1
∆K = Kf − Ki = 0 − m v 2
2
The work done by the net force is
W =−
1
kx m2 − µm g x m
2
Equating we have
1
1
m v 2 = k x m2 + µm g x m
2
2
Now µmg = 0.5 × 103 × 10 = 5 × 103 N (taking
g =10.0 m s -2). After rearranging the above
equation we obtain the following quadratic
equation in the unknown xm.
k x m2 + 2µm g x m − m v 2 = 0
1/2
− µ m g +  µ2m 2 g 2 + m k v 2 
xm =
k
where we take the positive square root since xm
is positive. Putting in numerical values we
obtain
xm = 1.35 m
which, as expected, is less than the result in
Example 6.8.
If the two forces on the body consist of a
conservative force Fc and a non-conservative
force Fnc , the conservation of mechanical energy
formula will have to be modified. By the WE
theorem
(Fc+ Fnc ) ∆x = ∆K
But
Fc ∆x = − ∆V
Hence,
∆(K + V) = Fnc ∆x
∆E = Fnc ∆x
where E is the total mechanical energy. Over
the path this assumes the form
Ef − Ei = Wnc
Where W nc is the total work done by the
non-conservative forces over the path. Note that
126
unlike the conservative force, Wnc depends on
the particular path i to f.
⊳
6.10 VARIOUS FORMS OF ENERGY : THE LAW
OF CONSERVATION OF ENERGY
In the previous section we have discussed
mechanical energy. We have seen that it can be
classified into two distinct categories : one based
on motion, namely kinetic energy; the other on
configuration (position), namely potential energy.
Energy comes in many a forms which transform
into one another in ways which may not often
be clear to us.
6.10.1 Heat
We have seen that the frictional force is not a
conservative force. However, work is associated
with the force of friction, Example 6.5. A block of
mass m sliding on a rough horizontal surface
with speed v0 comes to a halt over a distance x0.
The work done by the force of kinetic friction f
over x 0 is –f x0. By the work-energy theorem
m vo2/2 = f x 0 . If we confine our scope to
mechanics, we would say that the kinetic energy
of the block is ‘lost’ due to the frictional force.
On examination of the block and the table we
would detect a slight increase in their
temperatures. The work done by friction is not
‘lost’, but is transferred as heat energy. This
raises the internal energy of the block and the
table. In winter, in order to feel warm, we
generate heat by vigorously rubbing our palms
together. We shall see later that the internal
energy is associated with the ceaseless, often
random, motion of molecules. A quantitative idea
of the transfer of heat energy is obtained by
noting that 1 kg of water releases about 42000 J
of energy when it cools by10 °C.
6.10.2 Chemical Energy
One of the greatest technical achievements of
humankind occurred when we discovered how
to ignite and control fire. We learnt to rub two
flint stones together (mechanical energy), got
them to heat up and to ignite a heap of dry leaves
(chemical energy), which then provided
sustained warmth. A matchstick ignites into a
bright flame when struck against a specially
prepared chemical surface. The lighted
matchstick, when applied to a firecracker,
results in a spectacular display of sound and
light.
PHYSICS
Chemical energy arises from the fact that the
molecules participating in the chemical reaction
have different binding energies. A stable chemical
compound has less energy than the separated parts.
A chemical reaction is basically a rearrangement
of atoms. If the total energy of the reactants is more
than the products of the reaction, heat is released
and the reaction is said to be an exothermic
reaction. If the reverse is true, heat is absorbed and
the reaction is endothermic. Coal consists of
carbon and a kilogram of it when burnt releases
about 3 × 107 J of energy.
Chemical energy is associated with the forces
that give rise to the stability of substances. These
forces bind atoms into molecules, molecules into
polymeric chains, etc. The chemical energy
arising from the combustion of coal, cooking gas,
wood and petroleum is indispensable to our daily
existence.
6.10.3 Electrical Energy
The flow of electrical current causes bulbs to
glow, fans to rotate and bells to ring. There are
laws governing the attraction and repulsion of
charges and currents, which we shall learn
later. Energy is associated with an electric
current. An urban Indian household consumes
about 200 J of energy per second on an average.
6.10.4 The Equivalence of Mass and Energy
Till the end of the nineteenth century, physicists
believed that in every physical and chemical
process, the mass of an isolated system is
conserved. Matter might change its phase, e.g.
glacial ice could melt into a gushing stream, but
matter is neither created nor destroyed; Albert
Einstein (1879-1955) however, showed that mass
and energy are equivalent and are related by
the relation
E = m c2
(6.20)
where c, the speed of light in vacuum is
approximately 3 ×108 m s–1. Thus, a staggering
amount of energy is associated with a mere
kilogram of matter
E = 1× (3 ×108)2 J = 9 ×1016 J.
This is equivalent to the annual electrical output
of a large (3000 MW) power generating station.
6.10.5 Nuclear Energy
The most destructive weapons made by man, the
fission and fusion bombs are manifestations of
WORK, ENERGY AND POWER
127
Table 6.3 Approximate energy associated with various phenomena
nucleus like uranium
235 ,
92 U
is split by a neutron
into lighter nuclei. Once again the final mass is
less than the initial mass and the mass difference
translates into energy, which can be tapped to
provide electrical energy as in nuclear power
plants (controlled nuclear fission) or can be
employed in making nuclear weapons
(uncontrolled nuclear fission). Strictly, the energy
∆E released in a chemical reaction can also be
related to the mass defect ∆m = ∆E/c2. However,
for a chemical reaction, this mass defect is much
smaller than for a nuclear reaction. Table 6.3
lists the total energies for a variety of events and
phenomena.
⊳
the above equivalence of mass and energy [Eq.
(6.20)]. On the other hand the explanation of the
life-nourishing energy output of the sun is also
based on the above equation. In this case
effectively four light hydrogen nuclei fuse to form
a helium nucleus whose mass is less than the
sum of the masses of the reactants. This mass
difference, called the mass defect ∆m is the
source of energy (∆m)c2. In fission, a heavy
Example 6.10 Examine Tables 6.1-6.3
and express (a) The energy required to
break one bond in DNA in eV; (b) The
kinetic energy of an air molecule (10—21 J)
in eV; (c) The daily intake of a human adult
in kilocalories.
Answer (a) Energy required to break one bond
of DNA is
10 −20 J
~ 0.06 eV
1.6 × 10 −19 J/eV
Note 0.1 eV = 100 meV (100 millielectron volt).
(b) The kinetic energy of an air molecule is
10 −21 J
~ 0.0062 eV
1.6 × 10 −19 J/eV
This is the same as 6.2 meV.
(c) The average human consumption in a day is
107 J
4.2 × 103 J/kcal
~ 2400 kcal
128
PHYSICS
We point out a common misconception created
by newspapers and magazines. They mention
food values in calories and urge us to restrict
diet intake to below 2400 calories. What they
should be saying is kilocalories (kcal) and not
calories. A person consuming 2400 calories a
day will soon starve to death! 1 food calorie is
⊳
1 kcal.
The instantaneous power is defined as the
limiting value of the average power as time
interval approaches zero,
6.10.6 The Principle of Conservation of
Energy
The work dW done by a force F for a displacement
dr is dW = F.dr. The instantaneous power can
also be expressed as
We have seen that the total mechanical energy
of the system is conserved if the forces doing work
on it are conservative. If some of the forces
involved are non-conservative, part of the
mechanical energy may get transformed into
other forms such as heat, light and sound.
However, the total energy of an isolated system
does not change, as long as one accounts for all
forms of energy. Energy may be transformed from
one form to another but the total energy of an
isolated system remains constant. Energy can
neither be created, nor destroyed.
Since the universe as a whole may be viewed
as an isolated system, the total energy of the
universe is constant. If one part of the universe
loses energy, another part must gain an equal
amount of energy.
The principle of conservation of energy cannot
be proved. However, no violation of this principle
has been observed. The concept of conservation
and transformation of energy into various forms
links together various branches of physics,
chemistry and life sciences. It provides a
unifying, enduring element in our scientific
pursuits. From engineering point of view all
electronic, communication and mechanical
devices rely on some forms of energy
transformation.
6.11 POWER
Often it is interesting to know not only the work
done on an object, but also the rate at which
this work is done. We say a person is physically
fit if he not only climbs four floors of a building
but climbs them fast. Power is defined as the
time rate at which work is done or energy is
transferred.
The average power of a force is defined as the
ratio of the work, W, to the total time t taken
Pav =
P =
dW
dt
P = F.
W
t
(6.21)
dr
dt
= F.v
(6.22)
where v is the instantaneous velocity when the
force is F.
Power, like work and energy, is a scalar
quantity. Its dimensions are [ML2T –3]. In the SI,
its unit is called a watt (W). The watt is 1 J s–1.
The unit of power is named after James Watt,
one of the innovators of the steam engine in the
eighteenth century.
There is another unit of power, namely the
horse-power (hp)
1 hp = 746 W
This unit is still used to describe the output of
automobiles, motorbikes, etc.
We encounter the unit watt when we buy
electrical goods such as bulbs, heaters and
refrigerators. A 100 watt bulb which is on for 10
hours uses 1 kilowatt hour (kWh) of energy.
100 (watt) × 10 (hour)
= 1000 watt hour
=1 kilowatt hour (kWh)
= 103 (W) × 3600 (s)
= 3.6 × 106 J
Our electricity bills carry the energy
consumption in units of kWh. Note that kWh is
a unit of energy and not of power.
u Example 6.11 An elevator can carry a
maximum load of 1800 kg (elevator +
passengers) is moving up with a constant
speed of 2 m s–1. The frictional force opposing
the motion is 4000 N. Determine the
minimum power delivered by the motor to
the elevator in watts as well as in horse
power.
WORK, ENERGY AND POWER
129
Answer The downward force on the elevator is
F = m g + Ff = (1800 × 10) + 4000 = 22000 N
The motor must supply enough power to balance
this force. Hence,
P = F. v = 22000 × 2 = 44000 W = 59 hp
⊳
6.12 COLLISIONS
In physics we study motion (change in position).
At the same time, we try to discover physical
quantities, which do not change in a physical
process. The laws of momentum and energy
conservation are typical examples. In this
section we shall apply these laws to a commonly
encountered phenomena, namely collisions.
Several games such as billiards, marbles or
carrom involve collisions.We shall study the
collision of two masses in an idealised form.
Consider two masses m1 and m2. The particle
m1 is moving with speed v1i , the subscript ‘i’
implying initial. We can cosider m2 to be at rest.
No loss of generality is involved in making such
a selection. In this situation the mass m 1
collides with the stationary mass m2 and this
is depicted in Fig. 6.10.
by the second particle. F21 is likewise the force
exerted on the second particle by the first particle.
Now from Newton’s third law, F12 = − F21. This
implies
∆p1 + ∆p2 = 0
The above conclusion is true even though the
forces vary in a complex fashion during the
collision time ∆t. Since the third law is true at
every instant, the total impulse on the first object
is equal and opposite to that on the second.
On the other hand, the total kinetic energy of
the system is not necessarily conserved. The
impact and deformation during collision may
generate heat and sound. Part of the initial kinetic
energy is transformed into other forms of energy.
A useful way to visualise the deformation during
collision is in terms of a ‘compressed spring’. If
the ‘spring’ connecting the two masses regains
its original shape without loss in energy, then
the initial kinetic energy is equal to the final
kinetic energy but the kinetic energy during the
collision time ∆t is not constant. Such a collision
is called an elastic collision. On the other hand
the deformation may not be relieved and the two
bodies could move together after the collision. A
collision in which the two particles move together
after the collision is called a completely inelastic
collision. The intermediate case where the
deformation is partly relieved and some of the
initial kinetic energy is lost is more common and
is appropriately called an inelastic collision.
6.12.2 Collisions in One Dimension
Fig. 6.10 Collision of mass m1, with a stationary mass m2.
Consider first a completely inelastic collision
in one dimension. Then, in Fig. 6.10,
The masses m 1 and m 2 fly-off in different
directions. We shall see that there are
relationships, which connect the masses, the
velocities and the angles.
θ1 =θ2 = 0
m1v1i = (m1+m2)vf (momentum conservation)
vf =
6.12.1 Elastic and Inelastic Collisions
In all collisions the total linear momentum is
conserved; the initial momentum of the system
is equal to the final momentum of the system.
One can argue this as follows. When two objects
collide, the mutual impulsive forces acting over
the collision time ∆t cause a change in their
respective momenta :
∆p1 = F12 ∆t
∆p2 = F21 ∆t
where F12 is the force exerted on the first particle
m1
v1i
m1 + m 2
(6.23)
The loss in kinetic energy on collision is
∆K =
=
1
1
m1v1i2 − (m1 + m 2 )v 2f
2
2
1
1
m12
m1v12i −
v12i
2
2 m1 + m 2
=

m1 
1
m1v12i 1 −

2
m1 + m 2 

[using Eq. (6.23)]
130
PHYSICS
In performing an experiment on collision on a horizontal surface, we face three difficulties.
One, there will be friction and bodies will not travel with uniform velocities. Two, if two bodies
of different sizes collide on a table, it would be difficult to arrange them for a head-on collision
unless their centres of mass are at the same height above the surface. Three, it will be fairly
difficult to measure velocities of the two bodies just before and just after collision.
By performing this experiment in a vertical direction, all the three difficulties vanish. Take
two balls, one of which is heavier (basketball/football/volleyball) and the other lighter (tennis
ball/rubber ball/table tennis ball). First take only the heavier ball and drop it vertically from
some height, say 1 m. Note to which it rises. This gives the velocities near the floor or ground,
just before and just after the bounce (by using v 2 = 2 gh ). Hence you
will get the coefficient of restitution.
Now take the big ball and a small ball and hold them in your
hands one over the other, with the heavier ball below the lighter
one, as shown here. Drop them together, taking care that they remain
together while falling, and see what happens. You will find that the
heavier ball rises less than when it was dropped alone, while the
lighter one shoots up to about 3 m. With practice, you will be able to
hold the ball properly so that the lighter ball rises vertically up and
does not fly sideways. This is head-on collision.
You can try to find the best combination of balls which gives you
the best effect. You can measure the masses on a standard balance.
We leave it to you to think how you can determine the initial and
final velocities of the balls.
=
v1 f =
1 m1m 2 2
v1i
2 m1 + m 2
Consider next an elastic collision. Using the
above nomenclature with θ 1 = θ 2 = 0, the
momentum and kinetic energy conservation
equations are
m1v1i = m1v1f + m2v2f
(6.24)
m v =m v
(6.25)
2
1 1i
2
1 1f
+m v
2
2 2f
From Eqs. (6.24) and (6.25) it follows that,
v 2 f (v1i − v1 f ) = v12i − v12f
= (v1i − v1 f )(v1i + v1 f )
∴ v 2 f = v1i + v1 f
Substituting this in Eq. (6.24), we obtain
Case I : If the two masses are equal
v1f = 0
v2f = v1i
The first mass comes to rest and pushes off the
second mass with its initial speed on collision.
⊳
Hence,
v2 f =
Case II : If one mass dominates, e.g. m2 > > m1
v1f ~ − v1i
v2f ~ 0
The heavier mass is undisturbed while the
lighter mass reverses its velocity.
m 1v1i (v 2 f − v1i ) = m1v1 f (v 2 f − v1 f )
or,
(6.27)
2m1v1i
(6.28)
m1 + m 2
Thus, the ‘unknowns’ {v1f, v2f} are obtained in
terms of the ‘knowns’ {m1, m2, v1i}. Special cases
of our analysis are interesting.
and
which is a positive quantity as expected.
(m 1 − m 2 )
v1i
m1 + m 2
(6.26)
Example 6.12 Slowing down of neutrons:
In a nuclear reactor a neutron of high
speed (typically 107 m s–1) must be slowed
WORK, ENERGY AND POWER
131
to 103 m s–1 so that it can have a high
6.12.3 Collisions in Two Dimensions
probability of interacting with isotope 235
92 U
and causing it to fission. Show that a
neutron can lose most of its kinetic energy
in an elastic collision with a light nuclei
like deuterium or carbon which has a mass
of only a few times the neutron mass. The
material making up the light nuclei, usually
heavy water (D2O) or graphite, is called a
moderator.
Fig. 6.10 also depicts the collision of a moving
mass m1 with the stationary mass m2. Linear
momentum is conserved in such a collision.
Since momentum is a vector this implies three
equations for the three directions {x, y, z}.
Consider the plane determined by the final
velocity directions of m1 and m2 and choose it to
be the x-y plane. The conservation of the
z-component of the linear momentum implies
that the entire collision is in the x-y plane. The
x- and y-component equations are
Answer The initial kinetic energy of the neutron
is
K1i =
1
m1v12i
2
while its final kinetic energy from Eq. (6.27)
K1 f
1
= m 1v12f
2
2
 m − m2  2
1
= m1  1
v1i
2
 m1 + m 2 
K1i
m − m2 
= 1
 m 1 + m 2 
(6.30)
1
1
1
m1v1i 2 = m1v1 f 2 + m 2v2 f 2
2
2
2
(6.31)
2
while the fractional kinetic energy gained by the
moderating nuclei K2f /K1i is
f2 = 1 − f1 (elastic collision)
=
0 = m1v1f sin θ1 − m2v2f sin θ2
If, further the collision is elastic,
4m1m 2
(m1 + m 2 )2
We obtain an additional equation. That still
leaves us one equation short. At least one of
the four unknowns, say θ 1, must be made known
for the problem to be solvable. For example, θ1
can be determined by moving a detector in an
angular fashion from the x to the y axis. Given
{m1, m2, v1i , θ1} we can determine {v1f , v2f , θ2}
from Eqs. (6.29)-(6.31).
⊳
K1 f
(6.29)
One knows {m1, m2, v1i} in most situations. There
are thus four unknowns {v1f , v2f , θ1 and θ2}, and
only two equations. If θ 1 = θ 2 = 0, we regain
Eq. (6.24) for one dimensional collision.
The fractional kinetic energy lost is
f1 =
m1v1i = m1v1f cos θ 1 + m2v2f cos θ 2
One can also verify this result by substituting
from Eq. (6.28).
For deuterium m 2 = 2m 1 and we obtain
f1 = 1/9 while f2 = 8/9. Almost 90% of the
neutron’s energy is transferred to deuterium. For
carbon f1 = 71.6% and f2 = 28.4%. In practice,
however, this number is smaller since head-on
collisions are rare.
⊳
If the initial velocities and final velocities of
both the bodies are along the same straight line,
then it is called a one-dimensional collision, or
head-on collision. In the case of small spherical
bodies, this is possible if the direction of travel
of body 1 passes through the centre of body 2
which is at rest. In general, the collision is twodimensional, where the initial velocities and the
final velocities lie in a plane.
Example 6.13 Consider the collision
depicted in Fig. 6.10 to be between two
billiard balls with equal masses m1 = m2.
The first ball is called the cue while the
second ball is called the target. The
billiard player wants to ‘sink’ the target
ball in a corner pocket, which is at an
angle θ2 = 37°. Assume that the collision
is elastic and that friction and rotational
motion are not important. Obtain θ 1.
the masses are equal
v1i = v1f + v 2f
or
(
)(
v 1i 2 = v1 f + v 2 f ⋅ v1 f + v 2 f
= v1 f 2 + v 2 f 2 + 2v1 f .v2 f
)
132
=
PHYSICS
{v
2
1f
+ v 2 f 2 + 2v1 f v 2 f cos (θ1 + 37° )
}
(6.32)
Since the collision is elastic and m1 = m2 it follows
from conservation of kinetic energy that
v1i 2 = v1 f 2 + v 2 f 2
(6.33)
Comparing Eqs. (6.32) and (6.33), we get
cos (θ1 + 37°) = 0
or
θ1 + 37° = 90°
Thus, θ1 = 53°
This proves the following result : when two equal
masses undergo a glancing elastic collision with
one of them at rest, after the collision, they will
⊳
move at right angles to each other.
The matter simplifies greatly if we consider
spherical masses with smooth surfaces, and
assume that collision takes place only when the
bodies touch each other. This is what happens
in the games of marbles, carrom and billiards.
In our everyday world, collisions take place only
when two bodies touch each other. But consider
a comet coming from far distances to the sun, or
alpha particle coming towards a nucleus and
going away in some direction. Here we have to
deal with forces involving action at a distance.
Such an event is called scattering. The velocities
and directions in which the two particles go away
depend on their initial velocities as well as the
type of interaction between them, their masses,
shapes and sizes.
SUMMARY
1.
2.
3.
The work-energy theorem states that the change in kinetic energy of a body is the work
done by the net force on the body.
Kf - Ki = Wnet
A force is conservative if (i) work done by it on an object is path independent and
depends only on the end points {xi, xj}, or (ii) the work done by the force is zero for an
arbitrary closed path taken by the object such that it returns to its initial position.
For a conservative force in one dimension, we may define a potential energy function V(x)
such that
F (x ) = −
dV (x )
dx
xf
or
Vi − V f =
∫ F ( x ) dx
xi
4.
5.
6.
The principle of conservation of mechanical energy states that the total mechanical
energy of a body remains constant if the only forces that act on the body are conservative.
The gravitational potential energy of a particle of mass m at a height x about the earth’s
surface is
V(x) = m g x
where the variation of g with height is ignored.
The elastic potential energy of a spring of force constant k and extension x is
V (x ) =
7.
1
k x2
2
The scalar or dot product of two vectors A and B is written as A.B and is a scalar
quantity given by : A.B = AB cos θ, where θ is the angle between A and B. It can be
positive, negative or zero depending upon the value of θ. The scalar product of two
vectors can be interpreted as the product of magnitude of one vector and component
of the other vector along the first vector. For unit vectors :
ˆi ⋅ ˆi = ˆj ⋅ ˆj = k
ˆ ⋅k
ˆ = 1 and ˆi ⋅ ˆj = ˆj ⋅ k
ˆ =k
ˆ ⋅ ˆi = 0
Scalar products obey the commutative and the distributive laws.
WORK, ENERGY AND POWER
133
POINTS TO PONDER
1.
2.
3.
The phrase ‘calculate the work done’ is incomplete. We should refer (or imply
clearly by context) to the work done by a specific force or a group of forces on a
given body over a certain displacement.
Work done is a scalar quantity. It can be positive or negative unlike mass and
kinetic energy which are positive scalar quantities. The work done by the friction
or viscous force on a moving body is negative.
For two bodies, the sum of the mutual forces exerted between them is zero from
Newton’s Third Law,
F12 + F21 = 0
But the sum of the work done by the two forces need not always cancel, i.e.
W12 + W21 ≠ 0
4.
5.
6.
7.
8.
9.
However, it may sometimes be true.
The work done by a force can be calculated sometimes even if the exact nature of
the force is not known. This is clear from Example 6.2 where the WE theorem is
used in such a situation.
The WE theorem is not independent of Newton’s Second Law. The WE theorem
may be viewed as a scalar form of the Second Law. The principle of conservation
of mechanical energy may be viewed as a consequence of the WE theorem for
conservative forces.
The WE theorem holds in all inertial frames. It can also be extended to noninertial frames provided we include the pseudoforces in the calculation of the
net force acting on the body under consideration.
The potential energy of a body subjected to a conservative force is always
undetermined upto a constant. For example, the point where the potential
energy is zero is a matter of choice. For the gravitational potential energy mgh,
the zero of the potential energy is chosen to be the ground. For the spring
potential energy kx2/2 , the zero of the potential energy is the equilibrium position
of the oscillating mass.
Every force encountered in mechanics does not have an associated potential
energy. For example, work done by friction over a closed path is not zero and no
potential energy can be associated with friction.
During a collision : (a) the total linear momentum is conserved at each instant of
the collision ; (b) the kinetic energy conservation (even if the collision is elastic)
applies after the collision is over and does not hold at every instant of the collision.
In fact the two colliding objects are deformed and may be momentarily at rest
with respect to each other.
134
PHYSICS
EXERCISES
6.1
6.2
6.3
The sign of work done by a force on a body is important to understand. State carefully
if the following quantities are positive or negative:
(a) work done by a man in lifting a bucket out of a well by means of a rope tied to the
bucket.
(b) work done by gravitational force in the above case,
(c) work done by friction on a body
sliding down an inclined plane,
(d) work done by an applied force on
a body moving on a rough
horizontal plane with uniform
velocity,
(e) work done by the resistive force of
air on a vibrating pendulum in
bringing it to rest.
A body of mass 2 kg initially at rest
moves under the action of an applied
horizontal force of 7 N on a table with
coefficient of kinetic friction = 0.1.
Compute the
(a) work done by the applied force in
10 s,
(b) work done by friction in 10 s,
(c) work done by the net force on the
body in 10 s,
(d) change in kinetic energy of the
body in 10 s,
Given in Fig. 6.11 are examples of some
potential energy functions in one
dimension. The total energy of the
particle is indicated by a cross on the
ordinate axis. In each case, specify the
regions, if any, in which the particle
cannot be found for the given energy.
Also, indicate the minimum total
energy the particle must have in each
case. Think of simple physical contexts
for which these potential energy shapes
are relevant.
Fig. 6.11
WORK, ENERGY AND POWER
6.4
The potential energy function for a
particle executing linear simple
harmonic motion is given by V(x) =
kx2/2, where k is the force constant
of the oscillator. For k = 0.5 N m-1,
the graph of V(x) versus x is shown
in Fig. 6.12. Show that a particle of
total energy 1 J moving under this
potential must ‘turn back’ when it
reaches x = ± 2 m.
135
Fig. 6.12
6.5
(a) The casing of a rocket in flight
burns up due to friction. At
whose expense is the heat
energy required for burning
obtained? The rocket or the
atmosphere?
(b) Comets move around the sun
in highly elliptical orbits. The
gravitational force on the
Fig. 6.13
comet due to the sun is not
normal to the comet’s velocity
in general. Yet the work done by the gravitational force over every complete orbit
of the comet is zero. Why ?
(c) An artificial satellite orbiting the earth in very thin atmosphere loses its energy
gradually due to dissipation against atmospheric resistance, however small. Why
then does its speed increase progressively as it comes closer and closer to the earth ?
(d) In Fig. 6.13(i) the man walks 2 m carrying a mass of 15 kg on his hands. In Fig.
6.13(ii), he walks the same distance pulling the rope behind him. The rope goes
over a pulley, and a mass of 15 kg hangs at its other end. In which case is the work
done greater ?
6.6
Underline the correct alternative :
(a) When a conservative force does positive work on a body, the potential energy of
the body increases/decreases/remains unaltered.
(b) Work done by a body against friction always results in a loss of its kinetic/potential
energy.
(c) The rate of change of total momentum of a many-particle system is proportional
to the external force/sum of the internal forces on the system.
(d) In an inelastic collision of two bodies, the quantities which do not change after
the collision are the total kinetic energy/total linear momentum/total energy of
the system of two bodies.
6.7
State if each of the following statements is true or false. Give reasons for your answer.
(a) In an elastic collision of two bodies, the momentum and energy of each body is
conserved.
(b) Total energy of a system is always conserved, no matter what internal and external
forces on the body are present.
(c) Work done in the motion of a body over a closed loop is zero for every force in
nature.
(d) In an inelastic collision, the final kinetic energy is always less than the initial
kinetic energy of the system.
(a) In an elastic collision of two billiard balls, is the total kinetic energy conserved
during the short time of collision of the balls (i.e. when they are in contact) ?
(b) Is the total linear momentum conserved during the short time of an elastic collision
of two balls ?
6.8
136
PHYSICS
(c) What are the answers to (a) and (b) for an inelastic collision ?
(d) If the potential energy of two billiard balls depends only on the separation distance
between their centres, is the collision elastic or inelastic ? (Note, we are talking
here of potential energy corresponding to the force during collision, not gravitational
potential energy).
6.9 A body is initially at rest. It undergoes one-dimensional motion with constant
acceleration. The power delivered to it at time t is proportional to
(i) t1/2
(ii) t
(iii) t3/2
(iv) t2
6.10 A body is moving unidirectionally under the influence of a source of constant power.
Its displacement in time t is proportional to
(i) t1/2
(ii) t
(iii) t3/2
(iv) t2
6.11 A body constrained to move along the z-axis of a coordinate system is subject to a
constant force F given by
˜N
F = −˜i + 2 ˜j + 3 k
6.12
6.13
6.14
6.15
6.16
˜ are unit vectors along the x-, y- and z-axis of the system respectively.
where ˜i, ˜j, k
What is the work done by this force in moving the body a distance of 4 m along the
z-axis ?
An electron and a proton are detected in a cosmic ray experiment, the first with kinetic
energy 10 keV, and the second with 100 keV. Which is faster, the electron or the
proton ? Obtain the ratio of their speeds. (electron mass = 9.11×10-31 kg, proton mass
= 1.67×10–27 kg, 1 eV = 1.60 ×10–19 J).
A rain drop of radius 2 mm falls from a height of 500 m above the ground. It falls with
decreasing acceleration (due to viscous resistance of the air) until at half its original
height, it attains its maximum (terminal) speed, and moves with uniform speed
thereafter. What is the work done by the gravitational force on the drop in the first
and second half of its journey ? What is the work done by the resistive force in the
entire journey if its speed on reaching the ground is 10 m s–1 ?
A molecule in a gas container hits a horizontal wall with speed 200 m s–1 and angle 30°
with the normal, and rebounds with the same speed. Is momentum conserved in the
collision ? Is the collision elastic or inelastic ?
A pump on the ground floor of a building can pump up water to fill a tank of volume 30 m3
in 15 min. If the tank is 40 m above the ground, and the efficiency of the pump is 30%,
how much electric power is consumed by the pump ?
Two identical ball bearings in contact with each other and resting on a frictionless
table are hit head-on by another ball bearing of the same mass moving initially with a
speed V. If the collision is elastic, which of the following (Fig. 6.14) is a possible result
after collision ?
Fig. 6.14
WORK, ENERGY AND POWER
137
6.17 The bob A of a pendulum released from 30o to the
vertical hits another bob B of the same mass at rest
on a table as shown in Fig. 6.15. How high does
the bob A rise after the collision ? Neglect the size of
the bobs and assume the collision to be elastic.
6.18 The bob of a pendulum is released from a horizontal
position. If the length of the pendulum is 1.5 m,
what is the speed with which the bob arrives at the
lowermost point, given that it dissipated 5% of its
initial energy against air resistance ?
6.19 A trolley of mass 300 kg carrying a sandbag of 25 kg
is moving uniformly with a speed of 27 km/h on a
Fig. 6.15
frictionless track. After a while, sand starts leaking
out of a hole on the floor of the trolley at the rate of
0.05 kg s–1. What is the speed of the trolley after the entire sand bag is empty ?
6.20 A body of mass 0.5 kg travels in a straight line with velocity v =a x3/2 where a = 5 m–1/2 s–1.
What is the work done by the net force during its displacement from x = 0 to
x=2m?
6.21 The blades of a windmill sweep out a circle of area A. (a) If the wind flows at a
velocity v perpendicular to the circle, what is the mass of the air passing through it
in time t ? (b) What is the kinetic energy of the air ? (c) Assume that the windmill
converts 25% of the wind’s energy into electrical energy, and that A = 30 m2, v = 36
km/h and the density of air is 1.2 kg m–3. What is the electrical power produced ?
6.22 A person trying to lose weight (dieter) lifts a 10 kg mass, one thousand times, to a
height of 0.5 m each time. Assume that the potential energy lost each time she
lowers the mass is dissipated. (a) How much work does she do against the gravitational
force ? (b) Fat supplies 3.8 × 107J of energy per kilogram which is converted to
mechanical energy with a 20% efficiency rate. How much fat will the dieter use up?
6.23 A family uses 8 kW of power. (a) Direct solar energy is incident on the horizontal
surface at an average rate of 200 W per square meter. If 20% of this energy can be
converted to useful electrical energy, how large an area is needed to supply 8 kW?
(b) Compare this area to that of the roof of a typical house.
6.24 A bullet of mass 0.012 kg and horizontal speed 70 m s–1 strikes a block of wood of
mass 0.4 kg and instantly comes to rest with respect to the block. The block is
suspended from the ceiling by means of thin wires. Calculate the height to which
the block rises. Also, estimate the amount of heat produced in the block.
6.25 Two inclined frictionless tracks, one gradual and the other steep meet at A from
where two stones are allowed to slide down from rest, one on each track (Fig. 6.16).
Will the stones reach the bottom at the same time? Will they reach there with the
same speed? Explain. Given θ1 = 300, θ2 = 600, and h = 10 m, what are the speeds and
times taken by the two stones ?
Fig. 6.16
138
PHYSICS
6.26 A 1 kg block situated on a rough incline is connected to a spring of spring constant 100
N m–1 as shown in Fig. 6.17. The block is released from rest with the spring in the
unstretched position. The block moves 10 cm down the incline before coming to rest.
Find the coefficient of friction between the block and the incline. Assume that the
spring has a negligible mass and the pulley is frictionless.
Fig. 6.17
6.27 A bolt of mass 0.3 kg falls from the ceiling of an elevator moving down with an uniform
speed of 7 m s–1. It hits the floor of the elevator (length of the elevator = 3 m) and does
not rebound. What is the heat produced by the impact ? Would your answer be different
if the elevator were stationary ?
6.28 A trolley of mass 200 kg moves with a uniform speed of 36 km/h on a frictionless track.
A child of mass 20 kg runs on the trolley from one end to the other (10 m away) with a
speed of 4 m s–1 relative to the trolley in a direction opposite to the its motion, and
jumps out of the trolley. What is the final speed of the trolley ? How much has the
trolley moved from the time the child begins to run ?
6.29 Which of the following potential energy curves in Fig. 6.18 cannot possibly describe the
elastic collision of two billiard balls ? Here r is the distance between centres of the balls.
Fig. 6.18
6.30 Consider the decay of a free neutron at rest : n g p + e–
WORK, ENERGY AND POWER
Show that the two-body decay of this type must necessarily give an electron of fixed
energy and, therefore, cannot account for the observed continuous energy distribution
in the β-decay of a neutron or a nucleus (Fig. 6.19).
Fig. 6.19
[Note: The simple result of this exercise was one among the several arguments advanced by W.
Pauli to predict the existence of a third particle in the decay products of β-decay. This
particle is known as neutrino. We now know that it is a particle of intrinsic spin ½ (like
e—, p or n), but is neutral, and either massless or having an extremely small mass
(compared to the mass of electron) and which interacts very weakly with matter. The
correct decay process of neutron is : n g p + e – + ν ]
139
140
PHYSICS
APPENDIX 6.1 : POWER CONSUMPTION IN WALKING
The table below lists the approximate power expended by an adult human of mass 60 kg.
Table 6.4 Approximate power consumption
Mechanical work must not be confused with the everyday usage
of the term work. A woman standing with a very heavy load on
her head may get very tired. But no mechanical work is involved.
That is not to say that mechanical work cannot be estimated in
ordinary human activity.
Consider a person walking with constant speed v0. The mechanical work he does may be estimated simply
with the help of the work-energy theorem. Assume :
(a) The major work done in walking is due to the acceleration and deceleration of the legs with each stride
(See Fig. 6.20).
(b) Neglect air resistance.
(c) Neglect the small work done in lifting the legs against gravity.
(d) Neglect the swinging of hands etc. as is common in walking.
As we can see in Fig. 6.20, in each stride the leg is brought from rest to a speed, approximately equal to the
speed of walking, and then brought to rest again.
Fig. 6.20 An illustration of a single stride in walking. While the first leg is maximally off the round, the second leg
is on the ground and vice-versa
The work done by one leg in each stride is m l v 02 by the work-energy theorem. Here ml is the mass of the leg.
Note m l v 02 /2 energy is expended by one set of leg muscles to bring the foot from rest to speed v0 while an
additional m l v 02 /2 is expended by a complementary set of leg muscles to bring the foot to rest from speed v0.
Hence work done by both legs in one stride is (study Fig. 6.20 carefully)
Ws =2m l v02
(6.34)
Assuming ml = 10 kg and slow running of a nine-minute mile which translates to 3 m s-1 in SI units, we obtain
Ws = 180 J / stride
If we take a stride to be 2 m long, the person covers 1.5 strides per second at his speed of 3 m s-1. Thus the
power expended
J
stride
×1.5
stride
second
= 270 W
P = 180
We must bear in mind that this is a lower estimate since several avenues of power loss (e.g. swinging of hands,
air resistance etc.) have been ignored. The interesting point is that we did not worry about the forces involved.
The forces, mainly friction and those exerted on the leg by the muscles of the rest of the body, are hard to
estimate. Static friction does no work and we bypassed the impossible task of estimating the work done by the
muscles by taking recourse to the work-energy theorem. We can also see the advantage of a wheel. The wheel
permits smooth locomotion without the continual starting and stopping in mammalian locomotion.
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