Solved example: strength of femur

we have a numerical on the femur bone the femur is the Taiwan which is one of the strongest and the longest bone of our body it's given that the area of that area of cross-section is roughly around five centimeter square which means you're gonna assume it's pretty much like a cylinder all right and we're asked to find the maximum compressive load that it can withstand if stress shouldn't exceed 10 to the power 8 Pascal's all right so Lourdes is just a fancy term that we use for for weight in this context all right so we just have to calculate what's the maximum weight that we can put on this femur and make sure the stress does not exceed this number all right so I want you to pause the video and see if you can try this out yourself first all right so I have a femur bone drawn one more time over here and so we can assume that where we are putting C we're putting a YA force from here and maybe a force from here compressive load is something like this is just compressing it this will be the Lord right this will be the Lord and most of the time your thigh bone is getting compressed like this for example when you are standing the top part of the body is pushing down on you and the bottom maybe the ground or the bottom bone the bone that comes below this is pushing up on it all right so your born is in equilibrium because but it's it's under compression now due to these two forces you're born is going to get deform it's gonna get compressed so maybe something like this is going to happen you're gonna get compressed like this and therefore there will be a restoring force acting there'll be a restoring force acting on it and over here the restoring force is upwards over here the restoring force will be downwards let me just show you one of them and if you take a small piece of this bone now and zoom it in a small piece right at the top then notice there are two forces acting there's a force from the top that is the Lord that is the Lord and there's a force from the bottom that's the restoring force that's the restoring force and notice because the bone is in equilibrium this tiny piece is in equilibrium the force that is of the Lord exactly the same as the restoring force hope that makes sense right so all you do is figure out what's the maximum restoring force that can be generated inside this bone given that the stress shouldn't exceed 10 to the power 8 Pascal's so in this question we're not going to use Hookes law it's not needed because we don't care about strain there's no strain in the question at all so all we do is look at the stress so stress sigma is given to us that it shouldn't exceed 10 to the power 8 Pascal's and therefore the stress which is the restoring force divided by the area and area is given to us you can just plug in area 5 centimeter centi is 10 to the power minus 2 meters the whole square and that number shouldn't exceed 10 to the power 8 Pascal's which is just newton per meter squared now meter square and meter square cancels and so if you multiply by the area on both sides what you will end up with is the restoring force should be less than we multiplied by area on both sides so that gives us 5 times 10 to the power minus 2 square is minus 4 times times 10 to the power 8 Newton's or right over here or the force the restoring force has to be smaller than 5 times 10 to the power let's see minus 4 plus 8 is plus 4 10 to the power plus 4 Newtons and since we just saw that the restoring force is the same as the Lord we could go ahead and said that the force due to the Lord or the weight shouldn't exceed 5 50,000 Newtons and that's our answer so that's the maximum Lord the maximum load we can put is 50,000 Newtons now I just want you to think about the numbers over here 50,000 Newtons is roughly five thousand kilograms does that make sense because if you think in terms of weight fifty thousand weight is mg and so M has to be five thousand for that so you're born can withstand five thousand kilograms of weight and even if you if we could say will not go to five thousand because five thousand is like the limit right so let's let's go let's be safe let's go ten times smaller your your bone can withstand at least five hundred kilograms right ten times smaller yes and that's one single bone that means two bones together can withstand about a thousand kilograms I haven't that's incredible a thousand kilogram is like what 12 adults so what these numbers are telling you is that your bone is capable of carrying about 12 adults that's incredible that's amazing and that's why femur is the strongest one of the strongest bones of the human body but of course I don't encourage you to try this at home don't try to make a human pyramid at home because these are ideal conditions for example we are assuming that the Lord is acting along the axis that compression is going on if the two forces are not aligned over here then your bone can get twisted and when when when you have twisting forces with a different kind of stress and we will talk about that in future videos but anyways the don't try this at home but our bone is awesome