Modulus of elasticity - Class 11, Physics Video Lecture

in a previous video we saw that if you
have a wire or some elastic band say of
length L and some cross-sectional area a
and if we stretched it stressed it by
some length delta L then a restoring
force gets generated inside the wire
this force tries to restore the wire
back to its original shape that's what's
called a restoring force and from this
we define two new qualities one we call
as the stress which we define as the
restoring force per unit area stress is
the quantity that tells us how quickly
the material tries to snap back to its
original shape and the second quantity
was strain which is the change in length
per length or relative change in length
which tells us how much the material has
been deformed and if you're not familiar
with this or if you require a refresher
it would be great to go back and watch
that video and then come back over here
but what I'm going to do in this video
is to find the relationship between
stress and strain all right so let's do
that now one thing we could intuitively
say is that if you were to increase the
strain that means increase the amount of
deformation you produce then the stress
in the material would also increase
right I mean if you deform it more the
material try to snap back even more so
more strain should produce more stress
makes sense right and guess what
experiments supports that in fact
experiment tells us for small strain
values stress is even proportional to
strain what that means is if you have to
double the value of strain stress will
just get doubled linear proportionality
and this law which says stress is
proportional strain is called as Hookes
law Hookes law but this law has a
limitation remember that if you take a
spring and if you stretch it too much
then it will not come back to its
original shape it will have permanent
deformation so Hookes law is only valid
as long as you don't strain it too much
all right and so we say usually say this
only works within the elastic limits
within elastic limits and everybody will
have its own
elastic limit but there will be some
limit so as long as you're within that
this proportionality Horace and now we
can go ahead and replace this
proportionality with an equal to sign
and put a constant over here and what
we'll do is we'll try and understand
what this constant is really telling us
all right
so let's put some let's throw some
values of K and let's see what happens
to stress and strain if we start with
very low values of K so let's say let's
say K is very close to zero which means
very low value then notice that for a
given strain stress will also be very
close to zero right now what does that
mean well let's imagine if for have a
wire which k equal to zero and if you
strain it a little bit but you find that
the stress is zero that means if you let
go of it the wire doesn't come back to
its original shape does that make sense
which means that if K is close to zero
and the wire or the material has a very
low elastic property make sense right so
k equal to zero
very low values of K which means it just
means low elasticity low elasticity and
what if we have a very high value of K
what if K is very high what does that
mean well if K is very high then even
for a small amount of strain the stress
would be very high which means if you
take this material and strain even a
little bit the stress will be very high
and you'll snap back very quickly to its
original shape in other words it is very
highly elastic so high value of K means
high highly elastic so can you see that
K is actually telling us how elastic a
material is it is for that reason K is
called as the modulus of elasticity or
which is called as elastic modulus
elastic modulus and it just tells us how
elastic a material is higher the value
of K more elastic the material and
what's the units of K well since strain
has no units because it's length per
length it's any minimally cancels K
should have the same unit as stress and
the stress will have unit of
Newton's per meter square it's okay
we'll also have units of Newton's per
meter square and guess what there are
different ways in which you can deform a
material for example you can take a wire
and stretch it or you can even bend it
or you can twist it and so on and so for
different kinds of strain that you're
going to produce the poor material has
to produce an equivalent kind of
restoring force and so an equivalent
kind of stress so for every kind of
strain you produce there will be a
corresponding stress that will be
generated and for each of these stress
ten pairs
Hookes law works and so for each kinds
of stress and strain we will have
different kinds of elastic moduli all
right and now we'll talk about one
particular kind of stress and strain so
let me get rid of this and for whatever
follows please just remember Hookes law
that stress is proportional to strain
all right so the kind of stress and
strain we're going to talk about to
understand that let's let's look at an
example imagine we have a cylindrical
rod that is colored to a ceiling on one
end and it has a length L now what we
can do to this rod is we can start
pulling on it say we can pull on it this
way and when we do that the rod gets
stretched so Delta L is the amount by
which the rod is being stretched and
this produces a strain Delta L divided
by L and now the rod will try to undo
this by generating a restoring force to
bring it back to its original shape and
the restoring force now will be in this
direction and just like when you take a
string and you pull on it that we say
that the string is an attention
similarly the rod which is being pulled
now is under tension and so now if you
calculate this restoring force divided
by the cross-sectional area divided by
this cross-sectional area the resulting
stress now is called as tensile stress
so we call this as tensile stress and
this strain
Delta L by L we call that a stencil
strain now another thing you could do to
our rod so we take that same rod another
thing we could do to this is instead of
pulling we can now push on it so this
time let's push on it and when you do
that the rod now gets compressed so
again notice a strain is generated which
is Delta L by L and because of that now
the rod will try to undo that will try
to bring it back to equilibrium by
generating a restoring force in the
opposite direction this way again if we
calculate this stress now as the
restoring force divided by this cross
sectional area then we call this stress
as compressive stress compressive stress
again makes sense right because now the
rod is being compressed so compressive
stress and this stream that is resulting
strain over here is called as
compressive strain and together this
tensile and compressive we often call it
as longitudinal longitudinal stress or
strain and the word longitudal comes
from the fact that we are talking about
stresses and strains happening along the
axis of the rod so it's also called as
axial stress that's another name that we
do to this axial stress an axial strain
another thing to notice is that this
restoring force in both the cases is
perpendicular to the area can you see
that it's perpendicular to the area of
cross-section and therefore the
longitudinal stress is also sometimes
called as normal stress normal because
normal stands for perpendicular it's
telling us that the restoring force is
perpendicular to the cross-sectional
area there are some symbols used over
here so when you calculate the stress
the restoring force divided by the
cross-sectional area this longitudinal
stress or normal stress is usually
written by the symbol Sigma and
similarly if you take this stream that
is Delta L divided by L relative
the length remember then this strain is
usually represented by the symbol small
Greek letter e Epsilon it's called and
now if you use Hookes law we could say
that this stress is proportional is
proportional to strain proportional to
stream and so that proportionality can
be written as equal to and when elastic
modulus comes over here and this elastic
modulus is often written as Y and we
call that as Young's modulus Young's
modulus so the thing to note is whenever
we have the restoring force
perpendicular to the area we call that
as the normal stress or longitudinal
stress and whenever we're dealing with
longitudinal stress or strain the
modulus of elasticity that we need to
use is called as the Youngs modulus now
one last thing we'll do is look at a
couple of real-life examples where we
can see tensile and compressive stresses
well when it comes to tensile stress
this picture comes to my mind notice
that this is a this is a wrecking ball
which is used in demolition buildings so
it has a lot of weight and it's being
hung by some steel or steel wire or a
chain and notice the ball starts pulling
on this wire
which causes a tensile stress over here
so if you want to calculate how much
strain gets generated or something then
we are going to use Young's modulus over
here
similarly for compression the picture
that comes to my mind are pillars any
pillars that we use in any construction
notice that it's under compression
because the stuff that is coming on top
of it is pulling or it's pushing down on
this and the ground is pushing up on it
that's causing the pillar to be
compressed and due to this compression
there will be a stress and there will be
a strain and so again if you want to
calculate any of those then we are going
to use Young's modulus for that now in
general that Young's modulus for tensile
and compression need not be the same a
material could be more elastic when it
comes to tensile but less elastic when
it comes to compression and therefore
you might have two different values in
general for Young's modulus however in
any problem if they don't mention
two different values if it just give you
Young's modulus then you can use that
number for any one of them tensile or
compressive