Work is said to be done when the point of application of force has some displacement in the direction of the force.
The amount of work done is given by the dot product of force and displacement.
Work is independent of the time taken and is a scalar.
If the force and displacement are perpendicular to each other, then the work done is zero.
A person rowing a boat upstream is at rest with respect to an observer on the shore. According to the observer the person does not perform any work. However, the person performs work against the flow of water. If he stops rowing the boat, the boat moves in the direction of flow of water and work is performed by the force due to flow, as there is displacement in the direction of flow.
If the work is done by a uniformly varying force such as restoring force in a spring, then the work done is equal to the product of average force and displacement.
If the force is varying non - uniformly, then the work done
The area of F - s graph gives the work done.
SI unit of work is joule. Joule is the work done when a force of one newton displaces a body through one metre in the direction of force.
CGS unit of work is erg; 1 J = 107 ergs.
The work done in lifting an object of mass m through a height ‘h’ is equal to mgh.
When a body of mass m is raised from a height h1 to height h2, then the work done
= mg(h2 – h1).
Let a body be lifted through a height 'h' vertically upwards by a force 'F' acting upwards. Then, the work done by the resultant force is W = (F - mg)h.
The work done on a spring in stretching or compressing it through a distance x is given W = kx2 where k is the force constant or spring constant.
Work done in changing the elongation of a spring from x1 to x2 is
a) The work done in pulling the bob of a simple pendulum of length L through an angle as shown in the figure is
W=mgL(1 - cos) = 2mgLsin2(/2)
b) The velocity acquired by it when released from that position is
The work done in lifting a homogeneous metal rod lying on the ground such that it makes an angle '' with the horizontal, is W =
The work done in rotating a rod or bar of mass m through an angle θ about a point of suspension is W = (1 - cos) = mgLsin2(/2) where L is the distance of the centre of gravity from the point of suspension.
The work done in lifting a body of mass 'm' and density 'ds' in a liquid of density 'dl' through a height 'h' under gravity is
Work done in pulling back a part of length of a chain hanging from the edge onto a smooth horizontal table completely is W = .
i) Work done in moving a block of mass 'm' up a smooth inclined plane of inclination '' through a distance 's' is W = Fs = mg sin s
ii) If the plane is rough, then
W = mg (sin + μkcos)s
Work done by a gas during expansion at constant pressure 'P' is given by
Work done = (pressure) (change in volume)
W = P (dv) = P (V2 - V1)
Note: The above formula can also be used to calculate the work done by the heart in pumping the blood.
i) If pressure also varies then W =
Work is positive if V2 > V1 i.e. when gas expands and negative if V2 < V1 i.e when gas is compressed.
ii) Area of P - V graph gives work done by the gas. Rate of doing work is called power.
Power = = Force × velocity.
SI unit of power is watt and CGS unit is erg/second.
One horse power = 746 watt. If a vehicle travels with a speed of v overcoming a total resistance of F, then the power of the engine is given by P = .
If a body is rotated in circular path, the power exerted is given by P =
If a block of mass 'm' is pulled along the smooth inclined plane of angle '', with constant velocity 'v', then the power exerted is, p = (mg sin)v
If the block is pulled up a rough inclined plane then the power is P = mg (sin + μk cos)v
If the block is pulled down a rough inclined plane then the power is P = mg (sin - μk cos)v
When water is coming out from a house pipe of area of cross section 'A' with a velocity 'v' and hits a wall normally and
i) stops dead, then force exerted by the water on the wall is Av2ρ. And the power exerted by water is P = Av3ρ ( ρ = density of water)
ii) If water rebounds with same velocity (v) after striking the wall, P = 2Av3ρ
When sand drops from a stationary dropper at a rate of on to a conveyer belt moving with a constant velocity, then the extra force required to keep the belt moving with a constant speed
V is given by F = v. and the power required = P =
If a pump lifts the water from a well of depth 'h' and imparts some velocity 'v' to the water, then the power of pump
Power exerted by a machine gun which fires 'n' bullets in time 't' is P =
If a pump delivers V litres of water over a height of h metres in one minute, then the power of the engine (P) = .
A motor sends a liquid with a velocity 'V' in a tube of cross section 'A' and 'd' is the density of the liquid, then the power of the motor is
P = AdV3
The power of the lungs = K.E. of air blown per second.
= .(mass of air blown per second) × (velocity)2
The power of the heart = pressure × volume of blood pumped per second.
The capacity to do work is called energy. Work and energy have the same units.
Potential energy of a body or system is the capacity for doing work, which is possessed by the body or system by virtue of the relative positions of its parts.
Water stored in a dam, stretched rubber cord, wounded spring of a clock or toy etc., possess potential energy.
A wounded spring (such as in a clock or toy car) has potential energy U= K2 (K is a torque constant and is the number of radians through which it is wound).
Stretched or compressed spring possesses P.E.
a) Elastic potential energy of a stretched spring =
Where k = Force constant =
(S.I. unit of 'k' is Nm-1)
b) Work done in increasing the elongation of a spring from x1 and x2 is k()
The energy possessed by a body by virtue of its motion is called kinetic energy. It is measured by the amount of work which the body can do before coming to rest.
Running water, a released arrow, a bullet fired from a gun, blowing wind etc. possess kinetic energy.
If a body of mass m is moving with a velocity v, then its kinetic energy = mv2.
A flying bird possesses both K.E. and P.E.
The work done on a body at rest in order that it may acquire a certain velocity is a measure of its kinetic energy.
If the kinetic energy of a body of mass m is E and its momentum is P, then E = .
If the momentum of the body increased by ‘n’ times, K.E. increase by n2 times.
If the K.E. of the body increases by ‘n’ times, the momentum increases by times.
a) If the momentum of the body increases by p%, % increase in K.E.= p%
b) If the momentum of the body decreases by p%, % decrease in K.E.= p%.
a) If the K.E. of the body increases by e%, % increase in momentum= .
b) If the K.E. of the body decreases by e%, % decrease in momentum =
If two bodies, one heavier and the other lighter are moving with the same momentum, then the lighter body possesses greater kinetic energy.
If two bodies, one heavier and the other lighter have the same K.E. then the heavier body possesses greater momentum.
Two bodies, one is heavier and the other is lighter are moving with the same momentum. If they are stopped by the same retarding force, then
i) the distance travelled by the lighter body is greater. (s ∝ )
ii) They will come to rest within the same time interval.
Two bodies, one is heavier and the other is lighter are moving with same kinetic energy. If they are stopped by the same retarding force, then
i) The distance travelled by both the bodies are same.
ii) The time taken by the heavier body will be more (t ∝ ).
Two bodies, one is heavier and the other is lighter are moving with same velocity. If they are stopped by the same retarding force, then
i) The heavier body covers greater distance before coming to rest. (s ∝ m )
ii) The heavier body takes more time to come to test. (t ∝ m)
Simple pendulum : If the bob (mass m) of a pendulum of length (l) is raised to a vertical height (h) and then released, it executes SHM for smaller angles. The total energy is constant at all positions.
a) At the mean position, KE = mv2 (max), PE = 0(min)
b) At the extreme position, KE = 0 (min),
PE = mgl(1 - cos) (max)
c) KE at the mean position = PE at the extreme position
velocity at equilibrium position,
v = ,
d) When a pendulum of length l is held horizontal and relased.
Velocity at mean position, v =
e) The graphs for PE and KE are parabolic in shape.
Rebounding body :
f) If a body falling from height h1 loses x% of energy during the collision with the ground, the height to which it rebounds is
h2 h1 =
g) If a ball strikes a floor from a height h1 and rebounds to a height h2.
% loss of energy =
a) The PE at maximum height is maximum , PEH =
mgH = = mu2sin2 = E sin2
b) The KE at the highest point is minimum.
KEH = m(u cos)2 = mu2cos2 = Ecos2
c) Total energy = PEH + KEH =
d) The ratio of potential and kinetic energies of a projectile at the highest point is tan2.
RECOIL OF A GUN:
It a bullet of mass 'm' travelling with a muzzle velocity, is fired from a rifle of mass 'M', then
i) Velocity of recoil of the gun is V = mv/M
ii) KE of the bullet is greater than the KE of the rifle.
iv) when a gun of mass ‘M’ fire a bullet of mass ‘m’ releasing a total energy ‘E’.
Energy of bullet Eb =
Energy of gun Eg =
A block of mass 'M' is suspended by a string and a bullet of mass 'm' is fired into the block with a velocity 'v'. If the bullet embeds in the block, then
i) The common velocity of the system after the impact is V =
ii) The height to which it will rise is h =
Work–energy theorem : The work done by the resultant force acting on a body is equal to the change in its kinetic energy.
In general, the work done = change in energy.
Stopping distance of a vehicle is directly proportional to the square of its velocity and inversely proportional to the braking force.
If a body is thrown on a horizontal plane and comes to rest after travelling a distance 's', then
μ m g s = mv2
'μ' coefficient of friction
distance travelled before coming to rest
When a body of mass m falls freely from a height, its total energy is mgh.
When it falls through a distance x, its K.E. is mgx and P.E. is mg(h - x).
A stone of mass ‘m’ falls from a height ‘h’ and buries deep into sand through a depth ‘x’ before coming to rest. The average force of resistance offered by sand is
F = .
68. For a freely falling body or for a body thrown up K.E. at the ground is equal to the P.E. at the maximum height.
69. The total energy of a system is constant. Energy can neither be created nor destroyed. But it can be converted from one form to the other.
Examples on conversion of energy :
1. Electrical → Heat, e.g.: Iron, geyser over
2. Electrical → Light, e.g.: Filament bulb, Fluorescent tube
3. Electrical → Sound, e.g.: Loud speaker, Telephone receiver
4. Electrical → Mechanical. e.g.: Fan, Motor
5. Heat → Electrical. e.g.: Thermal power plant
6. Heat → Mechanical, e.g.: Steam locomotive
7. Mechanical → Electrical. e.g.: Dyano (Generator)
8. Sound → Electrical. e.g.: Microphone
9. Light → Electrical. e.g.: Photoelectric effect
10. Chemical → Electrical. e.g.: Primary cell
Rest mass energy : Every body or matter possesses a certain inherent amount of energy called rest energy even if it is at rest (so that K.E.= 0) and is not being acted on by a force (so that P.E.= 0). This rest mass energy is given by
E = mc2.