The terms work, power, energy are very familiar to us. We often use them in everyday language.
A student is reading his lesson,
A Lecturer is giving a lecture
A man is pushing against a wall...
In all the above three cases people felt that they are doing work. But actually their workdone is zero. In the above cases work means physical (or) mental labour.
But in physics work is done only when "a force produces motion"
Suppose Ramu has the capacity to work for 12 - 14 hours a day then he is said to have large "Stamina" (or) "Energy". Here these two words are closely related to "work". The capacity to do work is defined as energy. It is the root cause for each and every physical happening occur in the universe. The other word "power" is used in our daily life in different contexts with different meaning.
For example (i) A political party is in power (ii) Vijayendar singh gave a powerfull "Punch". Example (i) is not at all related to the scientific meaning of power. Example (ii) is some what nearer to the meaning power in physics. The aim of this chapter is to study the concept of these three (work, energy, power) physical quantities. These three quantities are scalar quantities.
What is a scalar quantity?
A physical quantity which has only magnitude but no direction is called a scalar quantity. e.g.: distance, speed, work, energy, power... etc. A Physical quantity which has both magnitude and direction is called a "vector quantity". e.g.: displacement, velocity, impulse ... etc. When any two vectors can be added or subtracted their resultant is also a vector. But when two vectors are multiplied the resultant may be in a vector form or scalar form. In this chapter we discuss about scalar product of two vectors.
Scalar Product (or) Dot Product:
If and are two non-zero vectors, the dot product between them is .
.
= AB cos θ, 'θ' is the angle between & . The dot product or scalar product of any two non-zero vectors is defined as the product of their magnitudes and cosine angle between them. The result of .. is a scalar.
* When displacement is produced in the direction of applied force then θ = 0°, cos 0° = 1
W = F S cos θ
Wmax = F S
Here the nature of workdone is +ve.
e.g.: 1) When a gas in a cylinder is compressed, the workdone by the compressive force is +ve.
2) When a person lifts a body from the ground, the workdone by the lifting force is +ve.
¤ If the force or its component is in the opposite direction to that of displacement, nature of workdone is −ve.
W = F S cos 180°
W = −F S
3) When breaks are applied on a moving vehicle, workdone by the breaking force is −ve, which is applied opposite to the motion of vehicle i.e., θ = 180°.
¤ Workdone becomes zero under three conditions.
(i) when the force acts perpendicular to the displacement, θ = 90°, cos 90° = 0.
i.e. W = F S cos 90º = 0
e.g.: When a man moves on a horizontal road with a load on his head.
(ii) If the displacement, S = 0, W = F(0) cos θ = 0
e.g.: A person tries to displace a wall by pushing it, if it does not move, W= 0.
(iii) Resultant force acting on the body becomes zero, F = 0
W = (0) s cos θ = 0
e.g.: If the body is moving with uniform velocity on a horizontal frictionless surface, W = 0.
Workdone by a force: work is said to be done by two types of forces.
(i) constant force (ii) variable force.
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Workdone by a variable force:
When the magnitude and direction of a force varies with position, the workdone by such
ENERGY
Energy of a body is its capacity to do work. Units and dimensional formula are same as that of work. It is a scalar quantity. Different forms of energy are mechanical, light, heat, electrical, nuclear, ........., etc. In this chapter we discuss only mechanical energy.
Example:
i) A bullet fired into a wall is able to penetrate into the wall by doing work against friction due to its energy.
(ii) Fast blowing wind is able to turn a wheel. Dimensional formula for energy is [ML2T−2], SI unit is joule (J), CGS unit is erg. 1 J = 107 erg. Other units are l ev = 1.6 × 10−19 J.
1 KWH = 36 × 1012 erg = 36 × 105 J.
In mechanics, mechanical energy is divided into two forms.
(i) Kinetic energy (ii) Potential energy.
Kinetic energy: It is the energy possessed by a body by virtue of its motion.
Examples: (i) A vehicle is in motion (ii) Water flowing along a river (iii) A bullet fired from a gun.
Consider a body of mass 'm' is moving with a velocity 'v'. A uniform force opposes its motion to bring the body to rest in a displacement 's'. The uniform retardation of the body due to force is 'a', which is obtained by the kinematic equation v2 − u2 = 2as.
Work - Energy theorem: This theorem gives the interrelation between work and KE of the body. 'The change in KE of the particle is equal to the work done on it by the net force'.
Consider a particle of mass 'm' is moving with an initial velocity 'u'. When it is under the action of a constant force 'F', it gets an acceleration 'a', velocity 'v' after a displacement 's'. Then v2 − u2 = 2as
On multiplying both sides with 
Work-Energy theorem for a variable force:
The time rate of change of kinetic energy is
Integrate on both sides from initial position xi to final position xf.
Thus, the 'Work Energy' theorem is proved for a variable force.
* This theorem is not only applicable for a single particle but also for a system.
* It is also applicable for a system under the action of variable force, conservative and non-conservative forces.
Potential Energy (PE):
The word PE suggests possibility (or) capacity for action. The word potential means 'stored'. Thus, the energy possessed by a body by virtue of its position is called 'PE'. It is defined for conservative forces. It does not exists for non-conservative forces.
Expression for Gravitational Potential Energy:
Consider a body of mass 'm' is on the ground. It is lifted vertically upward to a height of 'h'. If h < < < R (radius of earth), we can ignore the variation of 'g'. The gravitational force 'mg' is taken to be constant. Work done against gravitational force is 'mgh'. This work gets stored in the form of P.E. of the body. Now PE of the body is a function of 'h' denoted by U(h) = mgh -------- (1)
The gravitational force 'F' equal to -ve derivative of U(h) with respect to h.
The work done by a conservative force such as gravity depends on initial and final positions only. Dimensions of potential energy are [ML2T−2]. Unit in SI system is joule (J), 
same as kinetic energy and work.
Examples:
(i) Stone kept at certain height.
(ii) Energy possessed by a bended bow, wounded spring of a watch.
(iii) Energy possessed by water stored in a tank.
PE of a Spring:
A spring force is an example of a variable force which is conservative. The figure shows one end of massless spring attached to a rigid vertical support and other end to a block of mass 'm' resting on smooth horizontal surface. Let x = 0 denote the position of block, when spring is at its natural length. In an ideal spring, the spring force Fs is directly proportional to 'x'. When 'x' is the displacement of block from equilibrium position. 'x' and Fs may be +ve or − ve as shown in figure.
This force law for spring is called ''Hook's Law" which states that Fs = −kx, k - spring constant, unit is N − m−1. 'k' measures the stiffness of spring. When the block attached to spring is pulled out, the extension is xm, then work done by the spring force
Work done by spring force is always -ve. If block moves from initial position xi to final position xf, work done by spring force
If the block is pulled from initial position xi and returned to same position, then
Thus, work done by spring force depends on initial and final positions of the spring and independent on path followed. So, the spring force is a conservative force.
Conservative Force: Work done by the force around a closed path is zero and it is independent of the path, such a force is called conservative force.
Work done by the conservative force is stored in the form of potential energy.
e.g.: Gravitational force, electrostatic force and spring force.
Example: 1) In the absence of air resistance, a body is projected vertically up then work done by the gravitational force in moving the body through a height 'h' is W= −mgh and in return journey W = +mgh. On reaching the ground, the network done by the gravitational force in a round trip is zero.
WTotal = −mgh + mgh = 0
Example: 2) A body of mass 'm' lifted to a height 'h' from the ground level in different paths in between two points A & B. The work done by the gravitational force in all paths is same i.e., W1 = W2 = W3 = mgh. So, work done by the conservative force depends on the path followed by the body.
Non-Conservative force: If work done by the force around a closed path is not equal to zero and it is dependent on the path, such a force is called non-conservative force.
e.g.: Frictional force, viscous force.
Work done by a non-conservative force will not be stored in the form potential energy.
Example - 1: In the presence of air resistance, when a body is projected up, then it reaches a maximum height 'h'. Work done by the air friction in upward journey is '−fh' and in return journey 'fh'. On reaching the ground net workdone by the air friction is −ve, non - zero.
Example - 2: A block of mass 'm' is dragged on a rough horizontal surface through distance 's' from the the point 'P' to point 'Q' and then back to the point 'P'.
Work done by the frictional force from P to Q is −ve, Q to P is −ve. So the work done by the frictional force around a closed path is -ve and not equal to zero.
W1 ≠ W2 ≠ W3
Law of conservation of energy:
''Total mechanical energy of the system is conserved if the internal forces doing work on it are conservative and external forces do no work". If some of the forces are non-conservative, part of mechanical energy may get transferred into other forms such as heat, light and sound. The total energy of an isolated system remains constant. ''Energy neither be created nor be destroyed, but one form of energy transferred into another form of energy".
The principle of conservation of energy cannot be proved. However, no violation has been observed.
Total mechanical energy (E) = PE + KE = u + k = constant.
Law of conservation of energy in case of a freely falling body:
A body of mass 'm' is at a height 'h' from the ground as the body falls freely under gravity, the potential energy decreases and kinetic energy increases.
Total energy at Point A:
PE of the body = mgh
velocity of the body, v = 0
Total energy at Point A:
PE of the body = mgh
velocity of the body, v = 0
KE of the body = 
mv2 = 0
Total Energy (TE)A = PE + KE
= mgh + 0
= mgh ............... (1)
At point B:
As the body falls freely, consider a point 'B' in its path where AB = x
PE of the body = mg(h − x) = mgh − mgx
Let V1 be the velocity of the body at 'B'
Initial velocity u = 0 (at A),
Final velocity v = v1, acceleration a = g,
T.E.C = PE + KE = 0 + mgh = mgh ------ (3)
From (1), (2) & (3), TEA = TEB = TEC
Thus, law of conservation of energy is verified.
Various forms of energy:
Energy comes in many forms which transform into one form to another form. Few forms of energy are discussed below.
1. Heat: A block of mass 'm' sliding on a rough horizontal surface and comes to rest over a distance by work-energy theorem W= change of K.E., here in this case loss of KE by the block due to friction is transferred as heat energy. This raises the internal energy of the block. In winter, in order to feel warm, we generate heat by vigorously rubbing our palms together. A quantitative idea of transfer of heat energy is obtained by noting that 1 kg of water releases 42,000 J of energy when it cools by 10°C.
2. Chemical energy: 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 separated parts. If the total energy of the reactants is more than the product of the reaction, heat is released and reaction is said to be an exothermic reaction. If the reverse is true, heat is absorbed, and 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.
3. Electrical energy: The energy associated with an electric current is called electrical energy. The flow of electric current causes bulbs to glow, fans to rotate and bells to ring. An urban Indian household consumes about 200 J of energy per second on an average.
4. Nuclear energy: The most destructive weapons (fission and fusion bombs) made by man. Energy output of the sun and energy of stars is due to fusion of four light hydrogen nuclei form a helium nucleus whose mass is less than mass of reactants. This mass difference is called mass defect (∆m) which is the source of energy (E = ∆mc2). In fission, a heavy nucleus (92U235) splits into two smaller nuclies. The energy released in this reaction can also be related to the mass defect.
The Equivalence of Mass Energy:
Albert Einstein showed that mass and energy are equivalent and are related by the relation E = mc2, c is speed of light in vacuum ( = 3 × 108 m/s). The energy associated with a kg of matter.
E = (1) (3 × 108)2 = 9 × 1016 J
This is equivalent to the annual electrical output of a large (3000 MW) power generating station.
Collisions: Generally collision is refered to the case when two bodies come in physical contact with each other. Collision between two billiard balls or between two automobiles on a road, cricket bat hitting a ball are few examples of collision from every day life. In certain situations no physical contact but the path of one body is changed by the influence of another body, collision is said to have taken place.
Example - 1: In Rutherford's scattering, the α − particles are scattered due to electrostatic interaction between the α − particle and the nucleus from a distance, (no physical contact between α − particle & nucleus).
Example - 2: Two similarly charged particles separated by a finite distance may collide by interaction through their electric fields.
In a collision, before and after the impact, the interaction forces between the colliding particles becomes effectively zero.
In a collision the effect of external forces such as gravity (or) friction are not taken into account as due to small duration of collision. The average impulsive force is responsible for collision is much larger than external force acting on the system.
"The strong pyhysical interaction among bodies involing exchange of momenta in small interval of time is called a collision"
Classification of collisions on the basis of direction of motion of colliding bodies: On the basis of direction of motion of colliding bodies, collisions, are classified into two types.
1. Head on (or) one dimensional (or) direct collision
2. Oblique collision.
Head-on-collision: In a collision if the motion of colliding particles before and after collision are along same straight line. Then collision is said to be head on collision.
Oblique collision: If the motion of colliding objects before and after collision are not along the initial line of motion, such collisions are known as oblique collision. In this collision, if the objects travel in a plane before and after collsion, the collision is called two dimensional collision. If the objects travel in space and collide, it is called three dimensional collision.
Types of collisions on the basis of Law of conservation of kinetic Energy: On the basis of conservation of total kinetic energy of colliding objects, collisions are classified into two types. They are (a) Elastic collision (b) Inelastic collision.
Elasitc Collision: The collision, in which both momentum and kinetic energy remains constant are known as elastic collision.
e.g.: (i) Collsions between nuclei
(ii) Collisions between fundamental particles like electrons, protons, α − particles etc.
(iii) According to kinetic theory of gases, the collisions between gas molecules.
* In elastic collisions, objects regain their shape and size completely. Forces involved during interaction are conservation.
Inelastic Collions: The collisions in which kinetic energy is not conserved but law of conservation of momentum hold good are known as inelastic collisions.
Example: (i) Collision between cricket bal and bat.
(ii) Collision between automobiles on a road.
*In this collision the colliding bodies does not regain their shape and size completely after collision.
* Some fraction of mechanical energy is retained by colliding objects in the form of potential energy.
* Some (or) all the forces involved are non conservative in nature.
Perfectly inelastic collisions: If in a collsion, the colliding objects, stick together, and move with common velocity, then collision is called perfectly inelastic collision.
Example : (i) Collision between bullet and block of wood, when bullet is embedded in the block.
(ii) Collision between clay balls
(iii) Collision between positively and negatively charged particles.
In this collision law of conservation of linear momentum is conserved, kinetic energy is not conserved. But total energy is conserved.
Elastic collision in one dimension: Consider two masses m1, m2 suppose u1, u2, and v1, v2 be their velocities before collision and after collision respectively along the same straight line. Apply law of conservation of linear momentum, then momentum of the system before collision is equal to momentum of the system after collision. When two objects collide mutual impulsive forces acting over the collision time ∆t causes a change in their momenta.
i.e., ∆P1 = F12 ∆t
F12 is force exerted on the first particle by the second particle.
F21 is force exerted on the second particle by the first particle.
u1 + v1 = v2 + u2
(u1 − u2) = (v2 − v1) ............ (3)
Thus in head on collision (elastic), "The relative velocity of approach before collision is equal to the relative velocity of separation after collision".
From equation (3), v1 = v2 + u2 − u1
Substitute this value in equation (1), we get
m1[u1 − (v2 + u2 − u1)] = m2 (v2 − u2)
2m1u1 − m1v2 − m1v2 = m2v2 − m2u2
2m1u1 + (m2 − m1) u2 = (m1 + m2)v2
From equation (3), v2 = u1 − u2 + v1
Substitute this value in equation (1)
m1(u1 − v1) = m2 [(u1 − u2 + v1) − u2]
m1u1 − m1v1 = m2u1 − m2 u2 + m2v1 − m2u2
m1u1 − m2u1 + 2m2u2 = (m1 + m2)v1
1. State the law of conservation of energy and verify it in case of a body projected vertically upwards. Can we apply the law of conservation of energy to a system if the internal forces doing work are non-conservative?
A ball is projected vertically upwards from ground with an initial velocity of 9.8 ms-1. Find the maximum height reached by it using the law of conservation of energy.
Ans: Law of conservation of energy:
Statement: The total mechanical energy of a system is constant if the internal forces doing work on it are conservative and the external forces do no work.
Verification in case of vertically projected body:
Consider a body of mass 'm' projected vertically up from the ground with an initial velocity 'u'.
The total mechanical energy of the body is
E = K + U
where K = kinetic energy
U = Potential energy.
Let A, B, C be the three points on the ground, at heights h and H respectively.
At A: The kinetic energy of the body K = 
mu2
the potential energy of the body U = 0
The total mechanical energy of the body at A is
EA = K+U = mu2 + 0 = mu2
EA = mu2 ........................ (1)
At B: When the body reaches the point B, at a height 'h', let the velocity of the body be vB, then from A to B u = u , a = -g, s = h and v = vB = ?
using v2 - u2 = 2as
vB2 - u2 = 2(-g)h
vB2 = u2 - 2gh
vB = 
Now the kinetic energy of the body K = mvB2 = m (u2 - 2gh)
the potential energy of the body U = mgh
the total mechanical energy of the body at B is
EB = K + U = m (u2 - 2gh) + mgh = mu2
EB = mu2 ......................... (2)
At C: When the body reaches the maximum height position C, its velocity becomes zero.
The maximum height reached H = 
The kinetic energy of the body at C is K = 0
The potential energy of the body at C is U = mgH = mg (
) = mu2
The total mechanical energy of the body at C is
EC = K + U = 0 + mu2 = mu2
EC = mu2 ......................... (3)
From equations (1), (2) and (3), the total mechanical energy of the body remains constant under the action of gravitational force which is a conservative force.
Hence law of conservation of energy is verified in case of vertically projected body.
For non-conservative force:
We cannot apply the law of conservation of energy to a system if the internal forces doing work are non-conservative.
e.g.: In the case of a freely falling body and vertically projected body, if air resistance is considered which is a non-conservative force, the total mechanical energy is not constant.
Problem: Given: u = 9.8 ms-1, g = 9.8 ms-2, h = ?
By the law of conservation of energy mgh = mu2 
