WORK, ENERGY AND POWER
Basics of vector
DOT product: A.B = A B cos θ
Commutative law: A.B = B.A
Scalar product obeys the distributive law:
A. (B + C) = A.B + A.C A. (λ B) = λ (A.B)
__ _ _ _ _
i×I =j×j =k×k=1
_ _ _ _ _ _
i×j= j×k =k×i=0
WORK-ENERGY THEOREM
V2 - u2 = 2as where u and v are the initial and final speeds and s the distance traversed.
Multiplying both sides by m/2, we have
1/2mv2 -1/2 mu2 = mas = Fs
V2 - u2 = 2ad ( put s=d)
Once again multiplying both sides by m/2 ,
1/2mv2 -1/2 mu2 = ma.d = F.d
Kf - Ki = W
where Ki and Kf are respectively the initial and final kinetic energies of the object.
Work-Energy (WE) Theorem : The change in kinetic energy of a particle is equal to the
work done on it by the net force.
WORK
The work done by the force is defined to be the product of component of the force in the
direction of the displacement and the magnitude of this displacement. Thus
W = (F cos q) d = F.d
Dimensions, [ML2T-2]
The SI unit of work is joule (J)
If there is no displacement, there is no work done even if the force is large.
No work is done if :
d=0, s=0 and cos θ =900
A weightlifter holding a 150 kg mass steadily on his shoulder for 30s does no work on the load
during this time. (since d=0).
Block moving on a smooth horizontal table is not acted upon by a horizontal force (since there
is no friction), but may undergo a large displacement. (since force=0).
The force and displacement are mutually perpendicular.
EX. Block moving on a smooth horizontal table, the gravitational force mg does no work since
it acts at right angles to the displacement.
The moon’s orbits around the earth is perfectly circular then the earth’s gravitational force does
no work.
Work can be both positive and negative. If q is between 0 0 and 900, cos θ is positive.If q is
between 900 and 1800, cos θ is negative. In many examples the frictional force opposes
displacement and q = 180o. Then the work done by friction is negative (cos 180o = –1).
Energy
Energy is defined as the capacity or ability of a body to do work. Energy is scalar and its units
and dimensions are the same as that of work. Dimensions, [ML2T-2]
Thus, SI unit of energy is J. Some other commonly used units of energy are
KINETIC ENERGY
The kinetic energy of an object is a measure of the work an object can do by the virtue of its
motion. If an object of mass m has velocity v, its kinetic energy K is
Kinetic energy is a scalar quantity.
WORK DONE BY A VARIABLE FORCE
WORK-ENERGY THEOREM FOR A VARIABLE FORCE
It is an integral form of Newton’s second law. Newton’s second law is a relation between
acceleration and force at any instant of time. Work-energy theorem involves an integral over an
interval of time. Newton’s second law for two or three dimensions is in vector form whereas the
work-energy theorem is in scalar form
POTENTIAL ENERGY (Stored energy)
Potential energy is the energy stored in a body or a system by virtue of its position in a field of
force or due to its configuration. Potential energy is a scalar quantity
Dimensions, [ML2T-2] The SI unit of work is joule (J)
Ex.
A stretched bow-string possesses potential energy.
The earth’s crust is not uniform, but has discontinuities and dislocations that are called fault
lines. These fault lines in the earth’s crust are like ‘compressed springs’. They possess a large
amount of potential energy. An earthquake results when these fault lines readjust .
Gravitational Potential Energy
It is the energy associated with the state of separation between two bodies which interact via the
gravitational force. The gravitational potential energy of two particles of masses m1 and m2
separated by a distance r is
If a body of mass m is raised to a height h from the surface of the earth, the change in potential
energy of the system (earth+body) comes out to be
Thus, the potential energy of a body at height h, i.e. mgh is really the change in potential energy
of the system for h << R.
For the gravitational potential energy, the zero of the potential energy is chosen on the ground.
Electric Potential Energy: The electric potential energy of two point charges q1 and q2
separated by a distance r in vacuum is given by
Potential Energy of a Spring
Whenever an elastic body (say a spring) is either stretched or compressed, work is being done
against the elastic spring force. Fs = - kx
The work done is
The work done by the external pulling force F is positive since it overcomes the spring force.
W =1/2 kx2 where, k is spring constant and x is the displacement.
And elastic potential energy, U =1/2 kx2
If spring is stretched from initial position x1 to final position x2, then
Work Done = Increment in elastic potential energy = 1/2 k(x22-x12)
Spring force is a conservative force.
Total mechanical energy of the spring
The dimension of k/m is [T-2]
Principle of conservation of total mechanical energy
The total mechanical energy of a system is conserved if the forces, doing work on it, are
conservative.
The mechanical energy E of a system is the sum of its kinetic energy K and its potential energy
U.
E=K+U
If Wnet = 0 Mechanical energy, E = constant.
Conservation of energy: Energy may be transformed from one form to another but the total energy of
an isolated system remains constant. Energy can neither be created, nor destroyed.
Conservative force means
A force F(x) is conservative if it can be derived from a scalar quantity.
The work done by the conservative force depends only on the end points.
W = Kf – Ki = V (xi ) – V(xf )
. The work done by this force in a closed path is zero. ( ie xi =xf )
At the height H, the energy is purely potential. It is partially converted to kinetic at height h and
is fully kinetic at ground level. This illustrates the conservation of mechanical energy.
Remarks on conservative forces:
Information on time is absent.
Not all forces are conservative. Friction, for example, is a non-conservative force.
The zero of the potential energy is arbitrary. It is set according to convenience. For the
spring force we took V(x) = 0, at x = 0, i.e. the un stretched spring had zero potential
energy.
POWER:
It is a quantity that measures the rate at which work is done or energy is transformed.
Average power (P)avg =W/t
The shorter is the time taken by a person or a machine in performing a particular task, the larger
is the power of that person or machine.
Power is a scalar quantity and its SI unit is watt, where,
1W = 1 J/s
Instantaneous power,
Some other commonly used units of power are
