Work energy theorem - Work, Energy & Power Video Lecture - Class 11 | Best Video for Class 11

hello friends this video on work energy
and power part 7 is brought to you by
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please make sure that you have watched
all the videos from part 1 2 5 6 before
going ahead with part 7 now we would
study the physics behind or the physics
which relates kinetic energy good work
how is work related to kinetic energy we
have a very famous theorem called the
work-energy theorem which tells us the
relation between work and financing
energy so let us study the work energy
theorem
considering that the force is constant
that is a constant force is applied on a
body to produce displacement right here
we this is a pre assumption we have
assumed that the force does not change
with displacement force will remain the
same so what is work energy theorem it
states that work done on a body by a
force acting on it is equal to the
change in kinetic energy that means when
a force is applied on a body and there
is a displacement work is done right now
this work done on a body is equal to the
change in kinetic energy so basically
work done equals to change in kinetic
energy that is the statement of work
energy theorem now let us try to prove
the truth of this theorem now here we
have considered that the force which is
applied on the body is constant now let
us again remember our parametric
equations so from kinematics third
equation we know that V squared is equal
to u square plus 2ei s right now what we
can do is let us suppose that we
multiply both sides of the equation by
half M let us multiply both sides
by half em okay so what will they get
we'll get half MV squared is equal to
half M u square plus half M into 2 s so
this 2 + 2 will get cancelled so we get
1/2 MV squared is equal to half mu
square plus M a s now what is mass into
acceleration mass into acceleration is
equal to force so we can write it as
force into displacement what is force
into displacement this is nothing but
work done so we can write this + W so
what did we get we get work done is
equal to half MV square minus half em u
square now what is how can we spread
kinetic energy what is half mu square
here u is the initial velocity so this
is the initial kinetic energy and this
is the final kinetic energy so this is a
final minus initial so we see that the
work done on a body is equal to the
change in kinetic energy so this is how
we have proved or we have arrived at the
word energy theorem so work energy
theorem states that the work done on our
body is equal to the change in kinetic
energy here we are assuming that the
force applied on the body is constant
with time and is constant with
displacement so the force remains
constant now we will go ahead and we
will see the scenario when the force is
variable that means the force changes
with the displacement till now we saw
the scenario where the force remained
constant with displacement now we will
see where the force changes with
displacement let us now study the work
done by a variable force to understand
this latest draw the graph of force
displacement we considered we take force
along the y axis and displacement is
taken along x axis and this is our
origin we say that the force changes
with displacement that means the grout
will not be a straight line it would be
something like this later suppose the
force is changing the value of force is
changing at every point as the
displacement is changing so what we can
say from this we can say that force is
nothing but a function of displacement
course is a function of displacement so
when we write a quantity in this way it
means that force is a function of
displacement that has forced changes as
displacement changes let us say let us
denote this placement by X otherwise you
get country services displace okay now
let us suppose that at this point their
displacement was X 1 and at this point
the displacement is X 2 okay so we also
know that work done is equal to force
into displacement now here since force
is also changing at every point and
displacement is also changing so how
will we find the value of work done let
us consider a small section to make our
work easy let us suppose we consider a
small section we considered this much
section now we want to find out the work
done in this small section that means
the work done in moving from this point
to this point okay so what will be the
force for this let us suppose that this
distance this small distance is equal to
Delta X so we have to calculate the work
done in moving in covering a
displacement of Delta X so for the small
section the work done may also be very
small that is Delta W so Delta W will be
equal to f of X into Delta X because the
value of force will be
is variable it depends on X so we write
it as f of X into Delta X right
therefore the amount of work that the
total work done will be integration of
this small work done I hope you all know
what is integration integration means
integration means joining joining small
small parts to make a bigger part so we
have calculated the work done for this
small part now small small this many
parts together will constitute this
entire area I mean this entire length
path so the total work done will be
integration of this small work done
right so this will be equal to f of X
Delta X where X is changing from X is
equal to X 1 to X is equal to X 2 that
means we want to integrate this small
area from between these two points from
X 1 to X 2 so this will be the 1 so this
we can write as so this can also be
written as this is equal to summation of
f of X Delta X from X 1 to X 2 so both
of these are almost the same things
right so I hope you understood the
concept here I just wanted to make the
concept clear to you then the force is
variable that means when the force is
changing with displacement then we
denote force as f of X and in that case
the work done is given by the
integration of small amount of work done
for a small area right so this becomes W
is equal to integration of F X da F X
Delta X for X is equal to X 1 to X is
equal to X 2 now you might be wondering
why did I discuss this I discussed this
because till now we studied the
work-energy theorem for constant force
now we will see the modification in the
work-energy theorem for a variable force
so now you have clear what is a variable
force of force which changes with
displacement right so now let us go
ahead and see the work-energy theorem
for a variable force the statement of
the
energy theorem even in this case remains
the same what was the statement the
statement was work done on a body is
equal to the change in kinetic energy so
this statement will remain the same so
we will try to prove the work energy
theorem
considering that the force is variable
that is the force is changing with
displacement in this case let us start
we will track we will now prove this
work energy theorem now let us consider
we know that kinetic energy is equal to
half MV square right so now let us start
try to find what is change of kinetic
energy with respect to time because here
anyways we want change of kinetic energy
so we say DP by DT will be equal to half
M because this is constant into 2 v DV
by DT this 2 and 2 will get cancelled so
this will be equal to M V DV by DT okay
so here we can say this is equal to mV
what is DV by DT DV by DT is nothing but
acceleration so this will be equal to MV
a this we can write as M a into v what
is mass into acceleration force so this
is force into velocity right so what is
velocity now velocity is equal to rate
of change of displacement that is DX by
DT so we get DP by DT is equal to FD X
by DT now looking at both left-hand side
and right-hand side we can say DK is
equal to F D X now what is FD X if you
integrate both the sides let us suppose
I integrate both the sides you integrate
it from K initial to K final you
integrate this side from X initial 2x
final right that means for this
displacement now what will be this if
you
go to the previous slide and check you
will find that the work done in case of
a variable force is given by integration
of f into D X where X is equal to x1 to
x2 this is what we studied in our
previous slide so what is this this
right hand side term is nothing but work
done so this is equal to work done
now here DK k final to Kaisha liz k
final minus K initial and on the right
hand side we have W therefore we prove
our work energy theorem I hope it is
clear to you just that in both the work
energy theorem
remains true for both constant force and
variable force just that when we want to
prove the same we consider two different
approaches when we say it is a constant
force we can directly apply the
kinematics equation and prove it
when I say variable force in that case
variable force means we cannot say force
is equal to mass into acceleration
because the force is changing with
displacement it is not only dependent on
mass and acceleration it is also
dependent on displacement therefore in
this case what we have done is we have
said that the work done is equal to F T
X from X I to X F that means this is how
the work done is changing from initial
position to final position
so I hope it is clear do you know work
energy theorem so till now what all did
we study in this chapter what is word we
saw a numericals on work we start
studied at what is kinetic energy
relation between work and kinetic energy
now that we have studied kinetic energy
we will go ahead and study potential
energy see in this chapter basically in
energy we will study kinetic energy and
potential energy so let us now go ahead
with potential and thank you visit exam
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