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## Class 11 Physics (India)

### Unit 13: Lesson 1

Stress, strain, and modulus of elasticity

# Young's modulus of elasticity

In this video let's explore this thing called 'Young's modulus' which gives a relationship between the stress and strain for a given material. Created by Mahesh Shenoy.

## Video transcript

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