1. The law of conservation of energy states that energy cannot be created or destroyed, only changed from one form to another. The total energy in an isolated system remains constant. 2. The law of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it. 3. The law of conservation of angular momentum states that for a body or system of bodies with no external torque applied, the angular momentum about a fixed axis remains constant. Examples of this law include divers increasing their rotational speed during a spin and the evolution of stars from contracting gas clouds.

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Module # 12
Conservation of Energy & Momentum
Energy Conservation and Management
Energy conservation & management are the keys to using fuel
and electrical energy in most efficient way.
Law of Conservation of Energy
This law states that energy can neither be created nor destroyed.
This means that the total energy possessed by a body remains
constant. Thus the amount of energy in the universe is always the
same.
When a number of bodies are such that they can exert force upon
one another, but, no external force acts upon them, then, they are
said to form an isolated system of interacting bodies. In an
isolated system, the energy can be changed into different forms
but the total energy remains constant. It is possible to get only as
much energy out of a machine as we put into it. In practice, output
energy in a usable form is always somewhat less than the input.
This law when applied to gases means that the total heat supplied
or rejected in a system must be equal to the work done plus the
change in internal energy. Hence heat supplied to a system is
partly utilized in increasing its internal energy and partly utilized in

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doing some external work.
Examples
1 Consider a body of mass ‘m’ lying at height ‘h' above the
ground. As the body is at rest, therefore, its kinetic energy at 'h'
will be zero, but, potential energy at this point is mgh. The total
energy ‘E' at the height 'h' is
E = K.E. + P.E. = O + mgh = mgh _______ [1]
During its downward motion, its height from the ground
decreases. It, therefore, loses potential energy but its kinetic
energy increases at the same time as its velocity will go on
increasing. Suppose the body falls through a distance AB = x. Its
new height will be BC = (h - x).
To calculate kinetic energy at a point B, we find the velocity v by
using third equation of motion. Thus,
Initial velocity at A, Vi = 0
Distance covered, S = x
Acceleration, a = g
The velocity after covering distance x is Vf = V
So, according to third equation of motion,
Vf
2
- Vi
2
= 2aS

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OR
V2
– 0 = 2gx
OR
V2
= 2gx _____ [2]
Fig: (1) When a body falls from A to B, its P.E. is converted into
K.E.
Its kinetic energy at B is
K.E. = ½ mv2
By using [2], we get,
K.E. = mgx
Now, total energy at B is
E = K.E. + P.E. = mgx + mg (h-x) = mgh
This proves that the total energy of the body remains constant.

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During its downward, motion, the potential energy changes into
kinetic energy, but the sum of potential energy and kinetic energy
at the point remains constant.
2 Simple Pendulum
Now, we consider the example of a simple pendulum. A simple
pendulum consists of a small metallic bob suspended by a thin
but strong thread. If the bob is displaced from its mean position O
to a point A and then allowed to move, it starts vibrating about its
mean position. During its vibratory motion, the pendulum is at its
highest position at point A where its velocity is zero.
Therefore, at this point, the kinetic energy of pendulum is zero
while its potential energy is maximum. As the pendulum moves
back from the highest point A to the mean position O, it's height
starts decreasing and its velocity will go on increasing i.e., its
potential energy goes on decreasing, while, its kinetic energy
starts increasing. On reaching the mean position O, its height
reduces to zero and its velocity becomes maximum. At this point,
its potential energy is converted into kinetic energy i.e., K.E. of the

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bob is maximum while its P.E. is zero. The bob of the pendulum
moves from the mean position O towards the other extreme point
B due to inertia. During this motion, the pendulum gains height
and its velocity goes on decreasing. Its kinetic energy changes
into potential energy. On reaching the highest point B, the velocity
of the pendulum bob becomes zero. Here, its K.E. is zero while its
P.E. has become maximum. Now, bob starts moving back
towards the mean position O and the whole process is repeated.
As the bob repeats its vibratory motion, we get several successive
changes of K.E into P.E and vice-versa. However, the total
energy remains the same. Hence, the motion of the pendulum
supports the law of conservation of energy.
(3) When an electric current passes through a thin wire in a
bulb, it starts glowing producing heat and light, i.e. electric energy
is converted into heat and light energy.
(4) The chemical energy stored in food is converted into heat
energy as a result of digestion in the body. This energy keeps our
body warm and enables us to do work.
Law of Conservation of Momentum
When two or more bodies act upon one another, their total
momentum remains constant, provided no external forces are
acting.

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Rockets, which have made possible travelling in space, work on a
principle of physics called “law of conservation of momentum”.
The law of conservation of momentum is stated as "The
momentum of an isolated system (Isolated system is a system
which is not acted upon by an external force) always remains
constant"
Consider a system consisting of two balls of masses, m1 and m2
moving in a straight line, with velocities U1 and U2 respectively.
On colliding with each other they move with velocities V1 and V2
respectively.
The total momentum of the system before collision
= m1U1 + m2 U2
The total momentum of the system after collision
= m1V1 + m2V2
By the law of conservation of momentum,
m1U1 + m2U2 = m1V1 + m2V2.
When a body is moving over smooth horizontal surface with
uniform velocity, then, as the velocity of the body is constant, so,
its acceleration is zero and, therefore, it is not acted upon by an
unbalanced force i.e., F = 0 and a = 0. If we define the linear

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momentum of a body as the product of mass and velocity, then,
p= mv. The linear momentum of the object remains constant.
Conservation of Angular Momentum
The law of conservation of momentum in linear motion has a
counterpart in the rotational motion.
The law of conservation of angular momentum states that the
angular momentum about an axis of a given rotating body or
system of bodies is constant, if no external torque acts about that
axis.
The law of conservation of angular momentum has numerous
applications ranging from creation of stars down to subatomic
particles. Besides these, divers, ice skaters, ballet dancers and
others use this law to show spectacular feat. A few of the above
mentioned examples are discussed below.
(1) Diver Case
We know that L = mr2
. While diving, the diver gives a small
angular velocity to his body. When the body is curled, then, the
value of r is reduced and so the value of mr2
decreases. Hence,
the value of  must increase to keep the angular momentum (L)
constant and so, in this way, he is able to perform more
somersaults before striking the water. In this way, he can achieve

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very large rotational velocities and may appear blurred to the
observers.
(2) Evolution of Stars
Conservation of angular momentum plays very important role in
different natural phenomena. For instance, the evolution of stars
is considered to be a result of the law of conservation of angular
momentum. The stars are formed due to self-gravitation of clouds
of dust and gas in the interstellar space. These clouds of dust and
gas may initially possess small angular velocity due to the rotation
of the galaxy.
Due to gravitational effect, these clouds of dust and gas may
shrink to a dense object to conserve angular momentum, and so,
the angular velocity of such an object increases tremendously.
The process of gravitational contraction heats the stars to the
point/extant of luminescence and its internal temperature rises to
such an extent that a self-sustaining nuclear reaction starts.
The nuclear motion in some large stars became violently unstable
nearly at the end of evolution, thus ending in a supernova
explosion. The outer layer of the star is blown out in space,
leaving behind a very dense core where the nuclei come into
contact with each other, and thus, greatly reducing its mr2
and
increasing its angular velocity (). Such a supernova remnant in
the nearby crab Nebula rotates 30 times per second. Such stars
produce short pulses of radio and light and are called pulsars.