Work done by the force is equal to the product of the force and the displacement of the object in the direction of force.
If under a constant force F the object displaced through a distance s, then work done by the force
W = F * s = F s cos θ
where a is the smaller angle between F and s.
Work is a scalar quantity, Its S1 unit is joule and CGS unit is erg.
∴ 1 joule = 10^{7} erg
Its dimensional formula is [ML^{2}T^{-2}].
Work done by a force is zero, if
(a) body is not displaced actually, i.e., s = 0
(b) body is displaced perpendicular to the direction of force, i.e., θ = 90°
Work done by a force is positive if angle between F and s is acute angle.
Work done by a force is negative if angle between F and s is obtuse angle.
Work done by a constant force depends only on the initial and final Positions and not on the actual path followed between initial and final positions.
Unit: J (joule)
If force is along the line of motion, work done can be calculated by calculating area enclosed between F-S curve and displacement axis.
The time rate of work done by a body is called its power.
Power = Rate of doing work = Work done / Time taken
If under a constant force F a body is displaced through a distance s in time t, the power
p = W / t = F * s / t
But s / t = v ; uniform velocity with which body is displaced.
∴ P = F * v = F v cos θ
where θ is the smaller angle between F and v.
power is a scalar quantity. Its S1 unit is watt and its dimensional formula is [ML^{2}T^{-3}].
Its other units are kilowatt and horse power,
1 kilowatt = 1000 watt
1 horse power = 746 watt
Energy of a body is its capacity of doing work.
It is a scalar quantity.
Its SI unit is joule and CGS unit is erg. Its dimensional formula is [ML^{3}T^{-3}].
There are several types of energies, such as mechanical energy (kinetic energy and potential energy), chemical energy, light energy, heat energy, sound energy, nuclear energy, electric energy etc.
Mechanical Energy
The sum of kinetic and potential energies at any point remains constant throughout the motion. It does not depend upon time. This is known as law of conservation of mechanical energy.
Mechanical energy is of two types:
(a) Kinetic energy: The energy possessed by any object by virtue of its motion is called its kinetic energy.
Kinetic energy of an object is given by
k = 1 / 2 mv^{2} = p^{2} / 2m
where m = mass of the object, U = velocity of the object and p = mv = momentum of the object.
Unit of energy is the same as unit of work
(b)Potential energy: It is the energy of a body possessed by a body due to its position. Gravitational potential energy of a body of mass m at a height h from the ground or above a certain reference level is mgh.
There are three important types of potential energies:
(i) Gravitational Potential Energy If a body of mass m is raised through a height h against gravity, then its gravitational potential energy = mgh,
(ii) Elastic Potential Energy If a spring of spring constant k is stretched through a distance x. then elastic potential energy of the spring = 1 . 2 kx^{2}
The variation of potential energy with distance is shown in figure.
Potential energy is defined only for conservative forces. It does not exist for non-conservative forces.
Potential energy depends upon frame of reference.
(iii) Electric Potential Energy The electric potential energy of two point charges ql and q’l. separated by a distance r in vacuum is given by
U = 1 / 4πΣ_{0} * q_{1}q_{2} / r
Here 1 / 4πΣ_{0} = 9.0 * 10^{10} N-m^{2} / C^{2} constant.
Law of conservation of Energy: Energy can neither be created nor destroyed. Energy can be transformed from one form to another.
The work energy theorem: The work done by the resultant force acting on a body is equal to the change in its kinetic energy.
Conservative force: For conservative forces the sum of kinetic and potential energies of any object remains constant throughout the motion, while that by non-conservative force is path dependent.
Example of conservative force: Gravitational force and electric force
Example of non-conservative force: Force of kinetic friction
(i) Minimum value of velocity at the highest point is √gr
(ii) The minimum velocity at the bottom required to complete the circle v_{A} = √5gr
(iii) Velocity of the body when string is in horizontal position v_{B} = √3gr
(iv) Tension in the string
(v) When a vehicle is moving over a convex bridge, then at the maximum height, reaction (N_{1}) is N_{1} = mg – (mv^{2}/r)
(vi) When a vehicle is moving over a concave bridge, then at the lowest point, reaction (N_{2}) is N_{2} = mg + (mv^{2}/r)
(vii) When a car takes a turn, sometimes it overturns. During the overturning, it is the inner wheel which leaves the ground first.
(viii) A driver sees a child in front of him during driving a car, then it, better to apply brake suddenly rather than taking a sharp turn to avoid an accident.