S.-Serena-Work-power-energy.doc. this is useful to understand about work

CHAPTER 5
WORK ,ENERGY,AND POWER
by S.Serena steffani
XI C

INDEX
• Dot or Scalar Product – Properties of Scalar Product
• Work – Examples and Definition; Work done by Parallel Force and Oblique
Force
• When is Work Zero? Some Points about Work
• Positive, Zero and Negative Work – Examples
• Work done by a Variable Force
• Energy – Different Forms of Energy
• Kinetic Energy – Definition and Formula – Note on Kinetic Energy
• Work–Energy Theorem for Constant Force and Variable Force
• Potential Energy – Definition, Examples and Formula – Note on Potential
Energy
1

• Law of Conservation of Energy – Examples
• Conservative Force and Non-conservative Force
• Conservation of Mechanical Energy – Example using Gravitational Force &
Spring
• Collisions – Elastic, Completely Inelastic and Inelastic Collisions
• Elastic Collision in One Dimension and Inelastic Collision in One Dimension
• Elastic Collision in Two Dimensions
• Power and Note on Power
WORK
A horse pulling a cart does work
2

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
Work done by a parallel force
W = F s cos θ
= F s cos 0°
= F s
Work done by an oblique force:
W = (F cos θ) s
= F s cos
3

Work
 Work is a scalar quantity; work can be positive, zero or negative.
 Negative work does not mean it a vector quantity. Work done by friction is
negative as the displacement is opposite to the friction.
 Dimension of work is [ML2T-2].
 SI unit of work is ‘joule’ or ‘J’. 1 joule = 1 newton x 1 metre or 1 J =
1 Nm
 joule is the amount of work done when 1 newton of force acting on a body
displaces it through 1 metre.
 CGS unit of work is ‘erg’. 1 erg = 10-7 joule
4

 1 joule = 107 erg
 1 electron volt (eV) = 1.6 x 10-19 J
 1 calorie (cal) = 4.186 J
 1 kilowatt hour (kWh) = 3.6 x 106 J
Types of work
Positive, Zero and Negative Work done
The work done is positive if the force (or component of force) acts on a body
in the direction of its motion. (0º ≤ θ < 90º)
The work done is zero
if the force acts on
a body in the
direction
perpendicular to its
motion. (θ = 90º)
W = Fs cos θ
5

W = Fs cos 90º
W = 0 (cos 90º = 0)
The work done is negative if the force acts on a
body in the direction opposite to its motion. (90º
< θ ≤ 180º) W = Fs cos θ
W = Fs cos 180º
W = - Fs (cos 180º = -1)
Work done by variable force
More commonly, force is variable in practice.
The displacement covered by the body from xi to xf can be considered to be
made up of infinite number of infinitesimally small Δx.
Force F(x) may be considered to be constant over this Δx.
Then, the work done is
ΔW = F(x) Δx
The total work done is obtained by adding the rectangular areas of the
strips shown under the curve. W ≈ F X S
6

If the displacements are allowed to approach zero, then the summation approaches
a definite value equal to the area under the curve.
Then, the work done is
W = F(x) Δx
xi
Thus, for a varying force, the work done can be expressed as a definite integral of
force over displacement
What is energy
Energy is the ability to do work.
or
The amount of energy possessed by a body is equal to the amount of work it
can do when its energy is released.
Energy is a scalar quantity.
SI unit of energy is ‘joule’ or ‘J’.
CGS unit of energy is ‘erg’.
7

1 joule = 107 erg or 1 erg = 10-7 joule
different Forms of Energy
1. Mechanical energy ---> Potential energy and Kinetic energy
2. Heat energy 6. Light energy
3. Sound energy 7. Chemical energy
4. Electrical energy 8. Magnetic energy
5. Nuclear energy
Kinetic energy
• Kinetic energy is defined as the energy of a body by virtue of its motion.
• It is the measure of work a body can do by virtue of its motion.
• i.e. Every moving body possesses kinetic energy.
• Example:
• A moving cricket ball possesses kinetic energy.
8

Formula for Kinetic Energy
Suppose a body of mass ‘m’ moving with a velocity ‘u’ is acted upon by a force
‘F’ for time ‘t’. Let ‘v’ be the final velocity and ‘s’ be the displacement of the
body at the end of the time.
The work done by the force in displacing the body is given by
Work done = Force x
Displaceme
nt W = F x s
= ma x s
= m (as) ……………..2
= u2
+ 2as (1)
We know that v
or as = ½(v2
– u2
)
Substituting for as in (1)
W = m x ½(v2
– u2
)
9

W = ½m(v2
– u2
)
This work done is possessed by the body in the form of kinetic energy.
Therefore, KE = ½m(v2
– u2
)
If the body is initially at rest and its final velocity is ‘v’, then KE = ½mv2
Work – Energy Theorem
Suppose a body of mass ‘m’ moving with a velocity ‘u’ is acted upon by a force
‘F’ for time ‘t’. Let ‘v’ be the final velocity and ‘s’ be the displacement of the
body at the end of the time.
We know that v2
– u2
= 2as
Multiplying both the sides by m/2, we have
½mv2
– ½mu2
= mas
½mv2
– ½mu2
= F s
½mv2
– ½mu2
= W
Kf – Ki = W
The change in kinetic energy of a body is equal to the work done on it by the
net force.
The above equation can be generalised for 3-dimensions by employing vectors.
v2
– u2
= 2as becomes v2
– u2
= 2 a . s (since v . v = v2
and u . u = u2
)
10

Work–Energy Theorem For A Variable Force
Kinetic energy is given by K = ½ mv2
11
Differentiating w.r.t. ‘t’,
Integrating from initial position xi to final position xf, we have

Potential Energy
Potential energy is defined as the energy of a body by virtue of its position or
configuration.
Potential energy is stored in a compressed spring.
When the spring is released the potential energy stored in the spring does work
on the ball and the ball starts moving.
12
 F dx
xi
xf
 dK =
Ki
Kf
where Ki and Kf are initial and final kinetic energies
corresponding to xi and xf.
which is work–energy theorem for variable force.

1. Potential energy is path independent.
i.e. it depends on the net vertical displacement (height) of the body but
not on the path through which it is raised.
2. Potential energy in a round trip (i.e. over a closed path) is zero.
PE gained by the body = + mgh
PE lost by the body = - mgh
:
Total PE in round trip = + mgh – mgh = 0
Therefore, gravitational force is a conservative force.
3. If h is taken as variable, then the gravitational force F equals the negative
of the derivative of V(h) w.r.t. h. Thus, d
F = – mg
The negative sign indicates that the gravitational force is downward.
4. If the body with V(h) = mgh is released, the PE is converted into K = ½ mv2
.
v2
= 2gh
½ mv2
= mgh
5. The potential energy V(x) is defined if the force F(x) can be written as
13

Conservation Of Mechanical Energy
Suppose that a body undergoes displacement Δx under the action of a
conservative force F.
From work – energy theorem, we have
ΔK = F(x) Δx
If the force is conservative, the potential energy function V(x) can be defined
such that
– ΔV = F(x) Δx
The above equations imply that
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
14
This implies that
ΔK + ΔV = 0
Δ(K + V) = 0

Ki + V(xi) = Kf + V(xf)
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 total mechanical energy of a system is conserved if the forces, doing work
on it, are conservative.
If the block is moved from initial displacement xi to final displacement xf,
he work done by the spring force Fs is
15
Ws  kx dx
xi
xf
= – = ½ kxi
2
– ½ kxf
2
x = 0

Thus the work done by the spring force depends only on the end points. If
the block is pulled from xi and allowed to return to xi, then
Ws = ½ kxi
2
– ½ kxi
2
= 0
Since the work done in round trip is zero, the spring force is conservative.
We define the potential energy V(x) = 0 for a spring when the system of spring
and block is at equilibrium position.
For extension xe or compression xc
V(x) = ½ kx2
The above equation easily verifies that – dV/dx = – kx which is the spring force.
If the block of mass m is extended to xe and released from rest, then its total
mechanical energy at any arbitrary point x is the sum of its potential energy V(x) and
kinetic energy K.
½ kxe
2
= ½ kx2
+ ½ mv2
The above equation suggests that the speed and the kinetic energy will be
maximum at the equilibrium position x = 0.
16

i.e., ½ kxe
2
= ½ mvm
2
where xe is the maximum displacement and vm is
the maximum speed as given below.
Graphical representation of variation of PE and KE w.r.t. x’
and conservation of total ME of an oscillating spring:
– xe O xe x
The plots of PE and KE are parabolic.
They are complementary. i.e. one increases at the expense of the other.
17
vm =
k
m
xe
Energy
E = K + V

The total ME of the system remains constant at any arbitrary point
between – xe and xe.
Collisions
Elastic Collision
In elastic collision,
1. The total linear momentum is conserved,
2. The total kinetic energy of the system is also conserved and
3. After the collision, the bodies completely regain from their deformities.
Completely Inelastic Collision
In completely inelastic collision,
2. The total kinetic energy of the system is not conserved and
3. After the collision, the bodies stick together and move with common velocity.
Inelastic Collision
In inelastic collision,
18

2. The total kinetic energy of the system is not conserved and
3. After the collision, the bodies partly regain from their deformities and some of
he initial KE is lost.
19

Consider a body A of mass m1 moving with velocityu1 collides elastically with
another body B of mass m2 at rest. Let the bodies, after collision, move with
velocities v1 and v2 respectively along the directions as shown in the figure.
The collision takes place in two – dimensions, say in x-y plane.
Conservation laws have to be applied along x and y axes separately.
Along x – axis:
Kinetic energy is a scalar and hence, the law of conservation of energy is given
by
½ m1u12 = ½ m1v12 + ½ m2v22
We have four unknown quantities v1, v2, 1 and 2 and only 3 equations.
Therefore, atleast one more quantity must be known to solve the mathematical
problems.
20
Elastic Collision in Two Dimensions
Click to see the
collision…
v2
u1
m
1 m
2
m
2
v1
m
1

1

2 x
y
m
2v2 cos

2
m
1v1cos

1
m
1v1sin
1
m
2v2 sin
2
m1u1 = m1v1 cos 1 + m2v2 cos 2
Along y – axis:
0 = m1v1 sin 1 – m2v2 sin 2

Power
Power is defined as the time rate of doing work or consuming energy.
or
Power is defined as the rate of conversion of one form of energy into another
form of energy.
The instantaneous power is defined as the limiting value of the average power as
time interval approaches zero.
21
=
E
t
Pav=
W
t
Average Power =
Time taken
Work done
=
Time taken
Energy consumed
P =
dW
dt
The work done by a force F for a displacement dr is dW = F . dr
The instantaneous power can also be expressed as
P = F.
dr
dt

1. Power is a scalar quantity.
2. SI unit of power is ‘watt’.
3. 1 watt = 1 joule per second
4. 1 watt is the power when 1 joule of work is done in 1 second or 1 watt is the
power when 1 joule of energy is consumed in 1 second.
5. 1 kilowatt = 1000 watt or 1 kW = 1000 W
6. 1 megawatt = 1,000,000 watt or 1 MW = 106
W
7. Another unit of power is called ‘horse power’ or ‘hp’
8. 1 hp = 746 W
9. The power of engines of cars and other vehicles is measured by unit called
‘brake horse power’ which is equal to 1 horse power.
COMMERCIAL UNIT OF ENERGY
The commercial unit or trade unit of energy is kilowatt-hour (kWh).
1 kWh is the amount of electrical energy consumed when an electrical appliance
having a power rating of 1 kilowatt is used for 1 hour.
1 kilowatt-hour = 1000 watt x 3600 seconds = 3,600,000 Ws = 3.6 x 106
Joule
22

Conclusion
Work ,power energy , are intricately connected with each concept building
upon each other understanding these relationships is crucial for ;
1. EFFICIENT ENERGY USE ; minimizing energy waste and optimizing power
conceptions
2. INNOVATIVE TECNOLOGIES; developing sustainable solutions such as
renewable energy sources and energy effeciant machines
3. REAL WORLD APPLICATIONS ; informing fields like engineering ,physics,and
environmental science
The concept of work power and energy from fundamental foundation in
physics and engineering by grasping these principals, we can unlock a deeper
understanding of the world around us driving innovation and sustainability for
a bright future
23

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