Download 2. Work, Energy and Power

2. Work, Energy and Power
Work: Work is said to be done only when the force applied on a body makes the body
move ( ie there is a displacement )
 The amount of work depends on two factors:
(i) The magnitude and direction of force applied to an object
(ii) The distance/displacement through which the object moves
Measurement of work: The amount of work done by a force in moving a body is equal to the product of
the force and the displacement of the point of application of the force in the
direction of force.
Work = Force × {
F×S
Displacement of the point of application
}
of the force in the direction of force
ie W =
 Work is a scalar quantity.
 If either no force acts on a body or the body does not move, no work is done.
Expression of work ( W = F S 𝐜𝐨𝐬 𝜽 )
Let a constant force F is acts on body along AB and it displaces the body on an
inclined surface such that the point of application of the force moves from A to C. The
displacement of the body is S ( = S ) which is at an angle of inclination 𝜃 to the
direction of force.
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 1
W = F × AB
Work done
In ∆ ABC, cos 𝜃 =
AB
AC
=
AB
S
⇒ AB = S cos 𝜃
W = F × S 𝐜𝐨𝐬 𝜽
𝐃𝐢𝐬𝐩𝐥𝐚𝐜𝐞𝐦𝐞𝐧𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐩𝐨𝐢𝐧𝐭 𝐨𝐟 𝐚𝐩𝐩𝐥𝐢𝐜𝐚𝐭𝐢𝐨𝐧
Work = Force × {
}
𝐨𝐟 𝐭𝐡𝐞 𝐟𝐨𝐫𝐜𝐞 𝐢𝐧 𝐭𝐡𝐞 𝐝𝐢𝐫𝐞𝐜𝐭𝐢𝐨𝐧 𝐨𝐟 𝐟𝐨𝐫𝐜𝐞
The work done is equal to the product of
(i) Magnitude of force
(ii) Magnitude of displacement
(iii) Cosine of the angle 𝜃 between the directions of force F and displacement S
( ie cos 𝜃)
 Thus, the amount of work done is the product of force, displacement and the
cosine of the angle between the force and displacement.
 The work is a scalar quantity, in vector form W = →→
F. S
Special cases:
Case (i) If the displacement is in the direction of the force, i.e. 𝜃 = 0°, then the work
done is maximum and positive, Since cos 0° =1.
Example:
(1) In free fall of a body of mass m under gravity through a height h from A to B, the
force of gravity F ( = mg ) is in the direction of displacement S ( h ) and the work done
by the force of gravity is W = FS = mgh , Since cos 0° =1
(2) A coolie does work on the load when he rises it up against the force of gravity. The
force exerted by coolie (= mg) and displacement, both are in upward direction , Since
cos 0° =1.
Case (ii) If the displacement is normal to the direction of the force, i.e. 𝜃 = 90°, then
cos 90° =0 hence W = 0 Thus, work done is zero.
Example:
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 2
(1) When a coolie carrying some load on his head moves horizontally, no work is done
against the force of gravity because the displacement of load is normal to the direction
of force of gravity (which is vertically downwards)
(2) The coolie does work against the force of friction when he moves with the load.
(3) If the displacement is zero, then the work done is zero. This is the case when a
body is performing circular motion. Also direction of force and the direction of
displacement are in the direction mutually perpendicular to each other.
Conditions for work done by a force to be Zero.
(i) When there is no displacement ie S = 0.
(ii) When the displacement is normal to the direction of force ie 𝜃 = 90°.
Case (iii) If the displacement is in the direction opposite to that of the force, i.e. 𝜃 =
180° , then the work done is W = − F × S. This work is minimum and negative.
Example:
(1) When a body moves on a surface, the force of friction between the body and the
surface is in a direction opposite to the motion of the body. Therefore the force of
friction is negative.
(2) When a ball of mass m is thrown upwards from A to B to a height h, the
displacement h is opposite to the direction of force of gravity. So the work done by the
force of gravity is negative.
Work done by the body of gravity W = mgh
If a body of mass 𝑚 moves down from a height ℎ either directly or through an inclined
plane. The force of gravity or weight on the body is F = 𝑚𝑔 acting vertically
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 3
downwards and the displacement is in the direction of force is S = ℎ. Therefore the
work done by the force of gravity is
W = FS = 𝒎𝒈𝒉
 Note:- The work done by the force of gravity is the same whether a man comes
down from a certain height using a staircase or slope or a lift ( or elevator )
 Note:- If a boy of mass 𝑚 goes up through a vertical height either directly or
through a staircase or slope or lift, the work W = −𝑚𝑔ℎ is done by the gravity
on the body or the work W = 𝑚𝑔ℎ is done by the boy against the force of
gravity.
Units of work:S.I Unit:- The S.I unit of work is joule (J)
Since Work = force × displacement
∴
1 joule = 1 N × 1 metre
 One joule of work is said to be done when a force of one newton displaces the body
through a distance of one metre in its direction.
 The bigger units of work are kilo-joule ( kJ ), mega-joule ( MJ ) and giga-joule ( GJ
)
1 kJ = 𝟏𝟎𝟑 J
1 MJ = 𝟏𝟎𝟔 J
1 GJ = 𝟏𝟎𝟗 J
C.G.S unit:- The CGS unit of work is erg
∴
1 erg = 1 dyne × 1 cm.
 One erg of work is said to be done when a force of one dyne displaces the body
through a distance of one centimetre in its direction.
Relation between joule and erg
1 joule = 1 N × 1 m = 105 dyne × 102 cm
1 joule = 𝟏𝟎𝟕 erg
Power:- The rate of doing work is called power.
 Power is a scalar quantity.
Power =
𝐰𝐨𝐫𝐤 𝐝𝐨𝐧𝐞
P=
𝐖
𝐭𝐢𝐦𝐞 𝐭𝐚𝐤𝐞𝐧
𝐭
The power spent by a source depends on two factors:
(i) The amount of work done by the source and
(ii) The time taken by the source to do the work.
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 4
Note:- If a machine or a person does a given amount of work at a faster rate, more
power is spent by it ( or him ).
W=F×S
WKT
And
P=
W
t
=
F×S
𝑡
=F ×v
since
Power = force × average speed
×v
𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡
𝑡𝑖𝑚𝑒
or
= 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
P=F
If displacement S is at an angle 𝜃 with the direction of force F, then the power is
P = F v 𝐜𝐨𝐬 𝜽 = → . →
𝐅
𝒗
Units of power:S.I Unit:- The S.I unit of power is watt (W)
 If one joule of work is done in one second, the power spent is said to be 1 watt.
 Bigger units of power are kilo-watt ( kW ), mega-watt ( MW ) and giga-watt (
GW )
1 kW = 𝟏𝟎𝟑 W
1 MW = 𝟏𝟎𝟔 W
1 GW = 𝟏𝟎𝟗 W
 Smaller units of power are milliwatt ( mW ) and microwatt ( 𝜇W )
1 mW = 𝟏𝟎−𝟑 W
1 𝝁W = 𝟏𝟎−𝟔 W
C.G.S unit:- The CGS unit of work is erg per second
Relation between the S.I unit and C.G.S unit
1 W = 1 J𝐬 −𝟏 = 𝟏𝟎𝟕 erg 𝐬 −𝟏
Horse power:
1 H.P = 746 W = 0.746 kW
Differences between work and power
1. Work done by a force equal to the product 1. Power of a source is the rate of doing
of force and the displacement in the
work by it.
direction of force.
2. power spent depends on the time in
2. Work done does not depend on time
which work is done
3. S.I unit of power is watt (W)
3. The S.I unit of work is joule (j)
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 5
Energy:- The energy of a body is its capacity to do work.
 Energy is a scalar quantity.
Units of energy:



The SI unit of energy is joule (J), and its CGS unit is erg.
1 J = 107 erg.
Bigger units of energy are watt hour ( Wh )or kilowatt hour ( kWh ).
One watt hour is the energy spent by a source of power 1 W in 1 h.
ie 1 watt hour ( Wh )
= 1 watt × 1 hour = 1 J s −1 × 1 hour = 1 J s −1 × 3600
s = 3600 J
1 watt hour ( 1Wh ) = 3600 J = 3.6 kJ
 One kilowatt hour is the energy spent by a source of power 1 kW in 1 h.
1 kilo watt hour ( Wh ) = 1 kilowatt × 1 hour = 1000 J s −1 × 3600 s = 3.6 ×
106 = 3.6 MJ
1 kilowatt hour ( 1 kWh ) = 3.6
MJ
 Heat energy is usually is measured in calorie.
 One calorie is the heat energy required to raise the temperature of 1 g of water
through 1°C from 14.5 ℃ to 15.5℃.
1 J = 0.24 calorie or 1 calorie = 4·186 J or 4·2 J
 Kilocalorie is the bigger unit of energy of heat energy.
1 kilo-calorie (or 1 k.cal) = 4200 J
 The energy of atomic particles is measures in electron volt ( eV )
1 eV = 1.6 × 10−19 J
Differences between energy and power
1. The energy of a body is its capacity to 1.
Power of a source is the energy spent
do work.
by it in 1 S.
2. The energy spent does not depend on 2.
Power spent depends on the time in
time.
which energy spent.
3. S.I unit of energy is joule ( J ).
3.
S.I unit of power is watt ( W )
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 6
PART ( B )
Mechanical Energy
 The energy possessed by a body due to its state of rest or of motion is called
mechanical energy.
 The total mechanical energy of a body is equal to the sum of its kinetic energy and
potential energy.
Kinetic Energy
The energy possessed by a body by virtue of its state of motion is called kinetic energy.
It is denoted by ‘K’.
Qualitative value of kinetic energy:Suppose a body of mass m is moving with a velocity ‘v’. It is brought to rest by
applying a constant opposing force F. Let ‘a’ be the uniform retardation produced by
the force, and the body travels a distance ‘S’ before coming to rest.
Kinetic energy = Work done by retarding force in stopping it
= Retarding force × displacement
K = F × S = ma × S ----------------------------- (i)
Now to calculate the displacement S, we have
Initial velocity ( u ) = v , Final velocity ( v ) = 0
Since a is the retardation, so acceleration = −a
𝑣 2 = 𝑢2 + 2𝑎S ⇒ 0 = 𝑣 2 − 2𝑎S
From the equation,
∴ S=
∴ Eqn (i) becomes
K =
𝑣2
2𝑎
K = ma ×
𝟏
𝟐
𝒎𝒗𝟐
𝑣2
2𝑎
=
1
2
𝑚𝑣 2
𝟏
or kinetic energy = mass × (𝒗𝒆𝒍𝒐𝒄𝒊𝒕𝒚)𝟐
𝟐
Relation between kinetic energy and momentum:Let the body of mass m be moving with velocity v, then it’s KE=
and momentum (p) = mv
∴K=
1
2
𝑝 2
𝑚( )
𝑚
=
𝑝2
2𝑚
⇒
⇒v=
𝑝
1
2
𝑚𝑣 2
𝑚
𝑝2 = 2 m K
Thus kinetic energy and potential energy are related as
P = √𝟐𝒎𝐊
or K =
𝒑𝟐
𝟐𝒎
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 7
Work–Energy Theorem:According to the work–energy theorem “the work done by a force on a moving body is
equal to the increase in its kinetic energy”.
Let a body of mass m be moving with an initial velocity u. When a force F is
applied on the body along its direction of motion, it produces an acceleration a and
velocity of the body changes from u to v in moving a distance S. Then
Workdone by the force = Force × displacement
W=F× S
-------------------------- ( i )
2
2
From the equation
𝑣 = 𝑢 + 2𝑎S
-------------------------- ( ii )
Displacement
S=
𝑣 2 − 𝑢2
∴ Eqn (i) becomes W = ma ×
--------------------------- ( iii )
2𝑎
𝑣 2 − 𝑢2
2𝑎
1
1
2
2
= m ( 𝑣 2 − 𝑢2 ) =
1
m𝑣 2 − m 𝑢2
2
Thus the workdone on the body = Increase in kinetic energy
ie
W=
𝟏
𝟐
𝟏
m𝒗𝟐 − m 𝒖𝟐 = 𝑲𝒇 − 𝑲𝒊
𝟐
Forms of Kinetic Energy
 The motion of a body in a straight line path is called translational motion, and the
kinetic energy of the body due to motion in a straight line is called translational
kinetic energy.
 Eg: A car moving in a straight path, a fee falling body, a molecule of mono atomic
gas.
 When a body rotates about an axis, the motion is called rotational motion, and the
kinetic energy of the body due to rotational motion is called rotational kinetic
energy or simply rotational energy.
 Eg: A spinning top, a rotating wheel, the rotation of the earth on its own axis.
 When a body moves to and fro about its mean position, the motion is called
vibrational motion. The kinetic energy of the body due to its vibrational motion is
called vibrational kinetic energy or simply vibrational energy.
 Eg: A wire clamped at the both the ends and plucked at the middle, A steel strip
clamped at one end and vibrates when the free end is displaced slightly and then
released, atoms virate about their free end, a poly atomic molecule has the vibration
along with translational and rotational energies.
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 8
Potential Energy (U) :- The energy possessed by a body by virtue of its specific
position or changed configuration is called potential energy.
Eg:- A body placed at a height above the ground, a wound up watch spring, a
compressed spring, a bent bow, a stretched rubber etc.
Forms of potential energy:- Gravitational potential energy due to changed position
and elastic potential energy due to changed configuration.
Gravitational potential energy:- The potential energy possessed by a body due to its
position relative to the centre of the Earth is called its gravitational potential energy.
Larger the distance of the body from the earth greater its gravitational potential energy.
Elastic potential energy:- The potential energy possessed by a body in the deformed
state due to change in its configuration is called elastic potential energy. It is equal to
the amount of work done in deforming the body or in changing the configuration of the
body.
Gravitational Potential Energy:- The gravitational potential energy of a body at a
height above the ground is measured by the amount of work done in lifting it up to that
height against the force of gravity.
Let a body of mass m be lifted from the ground (or Earth surface) to a vertical height h.
The work W done on the body in lifting it to a height h is
W = Force of gravity (mg) x displacement (h) =mgh
Thus gravitational potential energy U = mgh
Thus, when a body is thrown vertically upwards, its potential energy increases.
Similarly, when a body is dropped from a height, its potential energy decreases.
* Potential energy changes into kinetic energy whenever it is put to use.
* When the string of a bow is pulled, some work is done which is stored in the
deformed state of the bow in the form of its elastic potential energy. On releasing the
string, the potential energy of the bow changes into the kinetic energy of the arrow
which helps to move it forward.
Distinctions between potential and kinetic energy
Potential energy
Kinetic energy
1. It is the energy possessed by a body
1. It is the energy possessed by a body
due to its changed position or
due to its state of motion.
configuration.
2. It is equal to the workdone that a
2. It is equal to the workdone in
moving body can do before coming
bringing the body to its changed
to rest.
state.
3. It can change into any other form.
3. It can change only in form of
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 9
kinetic energy
***************** THE REMAING CONCEPTS OF THIS PART AND ‘ C ’ PART
LEARN FROM THE TEXT BOOK ******************************
PART – D
Law of conservation of energy:- According to the law of conservation of energy,
energy can neither be created nor can it can be destroyed. Only it can change from one
state to the other.
Law of conservation of mechanical energy:- According to the Law of conservation
of mechanical energy whenever there is an interchange between the potential energy
and kinetic energy, the total mechanical energy remains constant when there is no
frictional forces.
ie K + U = Constant
Theoretical verification of K + U = constant for a freely falling body
Let a body of mass m be falling freely under gravity for a height h above the ground.
Let A, B and C are the different positions of the body during its free fall.
At the position A from a height h above the ground
Initial velocity of the body (u) = 0 since the body is at rest.
1
1
2
2
∴ Kinetic energy K = 𝑚𝑣 2 = 𝑚(0)2 = 0
Potential energy U = mgh
Hence total energy = K + U = 0 + mgh = mgh
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
----------------- (1)
Page 10
At the position B when it has fallen a distance x
Let 𝑣1 be the velocity acquired by the body at B after falling through a distance
x. Thus u=0 and s= x and a =g
From the Equation 𝑣 2 = 𝑢2 + 2𝑎𝑠
𝑣1 2 = 0 + 2𝑔 = 2𝑔𝑥
1
1
2
2
∴ Kinetic energy K = 𝑚 𝑣1 2 = 𝑚 × (2𝑔𝑥) = 𝑚𝑔𝑥
Now at B, height of a body above the ground = (h-x)
∴ potential energy U = mg (h-x)
Hence total energy = K + U = 𝑚𝑔𝑥+ mg (h-x)
= 𝑚𝑔𝑥 + mgh – mgx = mgh ----------------- (2)
At the position C on the ground when it has fallen a distance h
Let the velocity acquired by the body on reaching the ground be v. Then u=0,
s=h, a=g
From the equation 𝑣 2 = 𝑢2 + 2𝑎𝑠 = 0 + 2gh =2gh
Kinetic energy K =
1
2
1
𝑚𝑣 2 = 𝑚 × (2𝑔ℎ)= mgh
2
Now at C, height of a body above the ground = 0
Potential energy U = mgh= mg(0)=0
Hence the total energy = K + U = mgh + 0= mgh ------------------ (3)
Thus from the equations (1) , (2) and (3) mechanical energy ( ie K + U ) is constant
 Note:- As the body falls, its potential energy decreases and kinetic energy
increases. The potential energy changes into kinetic energy. Similarly when the
body is thrown up kinetic energy decreases and potential energy increases. The
kinetic energy changes into potential energy.
Conservation of mechanical energy in motion under gravity
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 11
Motion
Height above the
ground
Downward 𝒉 ( 𝒊𝒆 highest point A
motion
)
( ie free
𝟏
fall)
𝒉 ( 𝒊𝒆 middle point
𝟐
Kinetic
energy K
Potential
energy U
Total
energy
E=K+U
𝒎𝒈𝒉
0
𝒎𝒈𝒉
𝟏
𝒎𝒈𝒉
𝟐
𝟏
𝒎𝒈𝒉
𝟐
𝒎𝒈𝒉
0
𝒎𝒈𝒉
0
𝟏
𝒎𝒈𝒉
𝟐
𝒎𝒈𝒉
𝒎𝒈𝒉
𝒎𝒈𝒉
𝒎𝒈𝒉
𝒎𝒈𝒉
B
0 ( ie ground C )
Upward
motion
0 ( 𝒊𝒆 ground C )
𝟏
𝟐
𝒉 ( 𝒊𝒆 middle point
B
𝟏
𝒎𝒈𝒉
𝟐
0
𝒎𝒈𝒉
𝒉 ( 𝒊𝒆 highest point A
)
Table showing the kinetic energy and potential energy of a body in vertical
motion.
Application of Law of Conservation of Energy to a Simple Pendulum
 Let A be the resting position of the bob when it has potential energy. When the
bob is displaced to B from its resting position A, the bob gets raised by a vertical
height ℎ so the potential energy increases by 𝑚𝑔ℎ.
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 12
 Now on releasing the bob at B, it moves back to A. Its vertical height decreases
from ℎ to zero, So the potential energy decreases from 𝑚𝑔ℎ to zero and it gets
1
converted into kinetic energy ie 𝑚𝑣 2 = 𝑚𝑔ℎ.
2
 At the point B, the bob acquires a velocity 𝑣 = √2𝑔ℎ, so it moves towards C.
 At the point C, when the bob reaches to a vertical height ℎ above the point A,
again it acquires the potential energy 𝑚𝑔ℎ and its kinetic energy becomes zero.
So the bob momentarily comes to rest at the point C.
 But due to the force of gravity, the bob moves back from C to A.
Energy Degradation
* According to the law of conservation of energy, the total energy of the universe
remains constant.
* While transformation, the entire energy does not change into the desired form, but a
part of it always changes to some other form which is not useful.
* This conversion of energy to the undesirable (or non-useful) form is called
dissipation of energy.
* The gradual decrease of useful energy due to friction etc. is called degradation of
energy.
Examples:
* When we light a bulb by using electricity, less than 25% of electrical energy is
converted to light energy. The remaining energy changes into heat in the filament and
other invisible radiations.
* In transmission of electricity from the power generating station, a lot of electrical
energy is wasted in the form of heat energy in the line wires used for transmission.
* All machines have efficiency less than 1, which implies that only a fraction of input
energy is used for doing useful work and the rest of the input energy is wasted.
*When electrical appliances are run by electricity, the major part of electrical energy is
wasted in the form of heat etc….
**********************
Work, Energy and Power Notes
Grade X ICSE
ADVAITH PRAVEEN SHETTY
Page 13