SHM : Pendulum (Part - 21) - Oscillations, Physics, Class 11 Video Lecture

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video still part 20 before going ahead
with part 21 now we will talk about
certain simple system systems which
execute simple harmonic motion we will
talk about some two systems which are
very common examples of simple harmonic
motion so the first one is the simple
pendulum
even though till now since we started
this lesson I have always been taking
the example of a simple pendulum
explaining the various concepts so now
let us see specifically in the scenario
for a simple pendulum how do we find out
the time period or is there a specific
expression for time period of a simple
pendulum so first let us understand when
is a system for a simple pendulum it has
a simple pendulum actually mean a
pendulum or what is it like a simple
pendulum is a system which consists of a
mass of particle a particle of mass M
which is called the Bob of the pendulum
so you should have a mass of M which is
known as the Bob of the pendulum there
has to be an unstable massless string
there has to be a string which is
massless and which is unscriptural
people point what is people point the
point from which the thread is hanging
this point coverage the thread is
hanging has to be fixed so this point is
called P hold point and the equilibrium
position so the equilibrium position you
all know now right now the system is an
equilibrium position now when you take
this mass m to one extreme end and leave
it let us suppose if you take a very
thin almost massless string and you
attach a small ball at one end of it
you tie the other end of the string to
some support right and I leave it as it
is what will you see the mass will be
hanging vertically downwards and it will
be at rest now let us suppose you hold
the mass take it to one extreme end and
then leave it so what will happen after
that the mass will start oscillating it
will start oscillating like this so this
position where the mass tends to come
when it is left to its own is known as
the equilibrium position so what is a
simple pendulum a simple pendulum is a
system where you have a mass M attached
to a massless and answerable string from
a pivot point so any such system can be
called as a simple pendulum now we will
describe the motion of a simple pendulum
I mean how to rewrite the equation for a
simple pendulum or how does this motion
takes place
now when you observe this motion of the
pendulum you can see what are the
different forces that act on this Bob
right because there must be certain
forces as we already discussed before
there is something about restoring force
that is the restoring force the only
force which is acting on this wall no of
course the gravitational force cannot be
neglected whenever any object is hanging
this way they are asked to be a
gravitational force which is always
trying to pull the object towards itself
right so there is a gravitational force
that is FG which is acting downwards now
what is the gravitational force in this
case gravitational force is nothing but
the weight of the object which is mg
acting downwards right now what is the
other force the other force is the
tension which is acting on the ball
tension what is tension tension is the
force which acts due to the string and
this tension always acts in the opposite
direction of the gravitational force so
tension is due to the string and it acts
upwards now let us suppose the object is
at this position so the force mg will
act downwards right and the tension T
will act along this direction so the
tension always act towards the P volt
point it acts along the string towards
the P watch point so this is mg and this
is T later suppose this is the y-axis
we are considering the moment where the
body is making an angle theta with the
vertical so in this case we can resolve
this mg into two components like this
energy and would result into two
components if this angle is Theta this
angle will also be theta
so this will be so what will be the what
will be this component is mg that is mg
now if this angle is Theta what will be
the component along this axis this will
be nothing but F G cos theta and what
will be this this will be nothing but F
G sine theta I am just resolving into
which two components if this is F and
this happens theta G this will become F
cos theta and this will become F sine
theta right so we can see in our at
looking at this diagram we can see that
F G cos theta will be equal to G now
what is F G gravitational forces mg so
mg cos theta is equal to T now what
about mg sine theta mg sine theta is the
one which this please - there is no
motion along the vertical direction
right so the vertical forces that is
tension and F cos theta they will cancel
out each other they will both be equal
and opposite right so the forces acting
on the Y direction that is tension and
this they will both be equal and
opposite but what about this force that
is in G sine theta or
F G sine theta this force is the one
which produces the restoring torque
about the given point we spoke about
restoring force thread so this mg sine
theta so you have another force
component which is left out that is mg
this FG
it's nothing but mg sine theta so this
component is the one which provides the
least we'll talk about the people point
so because of this component the
movement of this verb is like that that
it always tries to come back to screen
position right so what is that what is
the direction of the restoring torque
now let us calculate the restoring
thought why is the term thought coming
into place here because thought comes
into K when we talk of a rotational
motion so in this case a rotation is
important about an axis this vertical
axis is the axis and there is a rotation
which is taking place so the restoring
dot now may be equal to force that is
the restoring force or the force mg sine
theta into L that is the length of the
arm let us suppose n is the length of
the pendulum so this becomes mg sine
theta into L and the direction is
negative so it is minus because it acts
in the opposite direction for
displacement Direction is opposite keep
displacement so this is the restoring
torque now in rotational motion we
studied that torque is equal to I into
alpha what is I I is moment of inertia
and alpha is angular acceleration right
so we can equate the previous equation
with this that is we can say I alpha is
equal to minus L mg sine theta so from
this we can say alpha is equal to minus
M G L sine theta divided by I now if if
theta is very very small if theta is
very small we can approximate sine theta
with theta right because or the script
sine theta the expansion of sine theta
they like theta plus Kita's cube by
factorial 3 and so on so when your
leader is very small you can neglect the
higher powers so you are left with keep
X we say but when theta is equal to
small sine theta can be written as theta
therefore alpha becomes minus MGL theta
divided by I so this shows that angular
acceleration is proportional with the
angular displacement this theta
represents the angular position or the
angular displacement so from here we
conclude that angular acceleration is
proportional this is angular
acceleration is proportional to the
angular displacement now using this we
can say whenever any quantity is
proportional to another quantity we can
convert the proportionality symbol into
an equal symbol by inserting a constant
so we can say alpha is equal to Omega
square theta what is omega R square
Omega square is nothing but this left
out quantity that is minus MGL by I so
we can say this implies Omega square is
equal to mg L by I now with this you can
say that Omega is equal to root over m
GL by I now what is omega by definition
we know that angular frequency is 2 pi
by T this is equal to root over mu G L
divided by I or we can say the time
period t is equal to 2 PI root over I
divided by M G and now what is I I is
moment of inertia now from our
rotational motion chapters we know that
moment of inertia about an axis is equal
to M R square that is
ask you to village square what does mass
air mass is the mass of the ball and
what is the radius radius is nothing but
the length of the pendulum so that is an
L square now we put this value of i we
get time period is equal to 2 PI root
over m l square divided by M G L so mm
will get answered L will get cancelled
so this comes out to be 2 PI root over
and L by G therefore time period is
equal to 2 PI root over L by G this is
an expression of time period for a
simple pendulum so in case of any simple
pendulum you can directly apply this
expression so time period of a simple
pendulum depends on the length of the
pendulum and acceleration due to gravity
so here we derive the expression for
time period of simple pendulum and by
doing so we also observed what are the
various forces responsible for this kind
of motion of the pendulum thank you
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