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Phys 1000F - Work, Energy and Power Notes
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This is a comprehensive and detailed note on Work, Energy and Power for Phys 1000F. Useful stuff!!
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Lesson-1WORK, ENERGY AND POWER
WORK
If a constant force F displaces a body through displacement s then the work done, W, is given by
d W Fscos θ F. s
Where s is magnitude of displacement and is angle between force and displacement
S.I. unit of work is Joule or Newton-metre.
If the work is done by a variable force F which displaces body from position A to B; having respective
position vectors r1 and r2 , the work WAB is given by
r2 F
WAB F.ds
= Area under the curve F-s graph.
r1
where ds is a small displacement of the body between A and B.
Work is a scalar quantity. If a number of forces are acting, the total work ds S
on the body is equal to the sum of the work done by all the forces.
SIGN CONVENTION OF WORK
When 0 < < 90°
F
direction of Motion
W = Fs cos is positive
i.e., when the force supports the motion of the body, work done by that force is said to positive
But when
2
F
direction of Motion
W = Fs cos = –ve
i.e. in this case force opposes the motion of the body and so work done by that force is said to be
negative.
, If a spring of spring constant k is stretched from initial extension xi to final extension xf, then the force
opposes extension such that F = –kx. The work dW done in extending the spring through dx is given by
dW = Fdx = -kx dx.
xf
Total work done W dW kxdx
xi
1 k x2f x2i
2
The work done by a spring force is negative. Force of friction also does some work which can be
negative, zero or positive depending upon whether the friction force is opposing motion, causing no
relative motion between surfaces and friction force is supporting the motion respectively.
when = then w = f.s cos 90° = 0 eg. an object moving on a circular path with constant velocity..
2
ENERGY
The energy of a body is the capacity of the body to do the work. It is a scalar quantity with the same unit
as that for work (Joule in S.I. unit). In mechanics, kinetic and potential energies are involved with the
dynamics of the body.
KINETIC ENERGY
It is the energy possessed by a body by virtue of its motion. A body of mass m moving with a velocity
1
v has a kinetic energy, E k m v .
2
2
since velocity is a relative parameter so KE is also a relative parameter. It is a scalar quantity Realtion
P2
between kinetic energy and momentum is given by K.E. = .
2m
TYPES OF FORCES
CONSERVATIVE AND NON-CONSERVATIVE FORCES
A force is said to be conservative if the work done by the force in moving a particle from one point to
another point does not depend upon the path taken but depends upon the initial and final positions. The
work done by a conservative force around a closed path is zero. Gravitational force, electric force,
spring force etc. are examples of conservative forces. All central forces are conservative forces.
If the work done by a force in moving a body from one point to another point depends upon the path
followed, then the force is said to be non-conservative. The work done by such a force in moving a
particle around a closed path is not zero. For example, the frictional forces and viscous forces works in
an irreversible manner and a part of energy is lost in overcoming these frictional forces (Mechanical
energy is converted to other energy forms such as heat, sound etc.). Therefore these are non-conservative
forces.
,POTENTIAL ENERGY
It is the energy of a body possesed by virtue of its position or the energy possessed by the body due to its
state.
It is independent of the way in which body is taken to this state. It is a relative parameter and depends
upon its value at reference level.
Change in potential energy can be defined as negative of work done by the conservative force in carrying
a body from reference position to the position under consideration
DEFINITION
i.e., U = – WAB
Where A is initial state, B is final state and WAB is the work done by conservative forces.
Since potential energy depends upon work done by conservative force only, hence it can’t be defined
for non conservative force(s) because in this type of force work done depends upon the path followed.
RELATIONSHIP BETWEEN FORCE AND POTENTIAL ENERGY
If a body is taken from A to B in such away that there is no net change in kinetic energy
work done= –change in P.E.
F r = U – (U + U)
= – U
U
Favg = – r
if r 0
U U
F = – rlim =–
0 r r
i.e. force at any point in the conservative field is equal to rate of change of potential energy at that
point. The above equation helps us to determine the nature of equilibrium at a point i.e., whether the
equilibrium is stable, unstable or neutral.
WORK - ENERGY THEOREM
The net work done by the resultant force acting on a particle is equal to the change in the kinetic energy
of the particle. If u and v are the initial and final speeds of a particle of mass m, the net work, Wnet,
done by the resultant force is given by
1 1
Wnet mv 2 mu 2
2 2
If k is change in the kinetic energy,,
k Wnet .
CONSERVATION OF MECHANICAL ENERGY
The sum of kinetic and potential energy in conservative systems is constant but it can be transformed
from one form to another form. The sum of total change in potential energy U and total change in
, kinetic energy k is zero if only conservative forces are acting on the system and there is no loss of
energy in overcoming non-conservative forces.
U + k = (U + k) = 0
Integrating, U + k = constant.
Thus if only conservative forces act and perform work, the total mechanical energy of the system is
conserved i.e. the change in the total mechanical energy of the system is zero. This is called conservation
of mechanical energy.
1. Draw a sketch of the physical situation for the initial and final positions by choosing a suitable coordinate
system.
2. Specify the reference level for the potential energy and correct direction of motion of body/bodies. Any
convenient level can be chosen like zero gravitational potential energy at ground level. However potential
energy for the unstretched spring of natural length is necessary zero in every case.
3. Write the sum of kinetic energy, gravitational/electrostatic/magnetic potential energy and spring potential
energy in compressed or stretched state for the respective initial and final position of the body.
4. Equate the sum of the initial energies with the sum of final energies in the case of conservative forces.
However additional contribution by irreversible frictional forces should be taken into account for such
non-conservative forces as may be present so that total energy is constant in the initial and final states.
5. Solve the unknown quantity by using the process of elimination.
POWER
It is defined as the rate at which the work is done. If an amount of work W is done in time t ,
W
average power, Pav
t
W dW
Instantaneous power P Lim
t 0
t dt
Work done by a force F on an object that has infinitesimally small displacement ds is dW = F ds.
then instantaneous power, P d W F ds F . v
dt dt
S.I. unit of power is watt or Joule/second and it is a scalar quantity.
MOTION IN A VERTICAL CIRCLE
Consider a particle of mass m attached to one end of a string and rotated in a vertical circle of radius r
with centre O. The speed of the particle will decrease as the particle travels from the lowest point to the
highest point and the speed will increase when the particle travels from the highest point to the lowest
point due to acceleration due to gravity.
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