How tall can mountains get?

when I think of the word tall one of the things that comes to my mind our mountains mountains are pretty tall but here's a question how high can they get do you think they can be as tall as they want there's nothing limiting that or do you think something might limit the maximum height a mountain can ever achieve to answer this question we need to focus not at the top of the mountain but at the bottom of the mountain so if we take say a small chunk of the mountain so if we take if we zoom in over here and concentrate on a small chunk of a mountain say we look at a rock then this rock is being pressed by the weight of the mountain on top of it what happens to a rock when you press it well when we press it in our hands since the forces are very tiny it feels like the rock is not being deformed but the rock gets deform microscopic deformations and under such horrendous forces the forces over here is large forces over here the deformations are quite significant and we have studied a lot about what happens when you deform materials when you strain them we've seen that within elastic limits the material just snaps back but if the strain is very high then the material won't snap back anymore instead it'll start flowing like a liquid and we've talked a lot about this when we when we spoke about the stress-strain graph in previous videos so if you need a refresher it would be a great idea to go back and watch that and then come back over here but the important thing over here is that if the forces so there are forces acting on this rock from all directions and if this forces go beyond the elastic limits then this rock no longer maintains its shape it starts flowing like a fluid and as a result if the bottom part of this of the mountain starts flowing or starts behaving like a liquid it'll start flowing away like this and the mountain will start sinking and it's for that reason there must be a maximum height above which the motor will just start sinking under its own weight so yeah a maximum height exists now can we calculate that that seems like a very challenging question I mean there could be so many details involved what kind of mountain were dealing with what kind of stress we are dealing with and all that stuff however we can sort of make an order of magnitude magnitude estimate based on what we've just learned so if you look at rocks like this it turns out that they're elastic limit elastic limit is roughly roughly around 3 times 10 to the power 8 Newton's per meter square and this is like an average value of the stuff that usually mountains are made of so again it's very close to that and so we know that if the pressure over here exceeds this value and we can pretty much say our mountain is going to sing down so all we do know now is figure out what is the pressure over here again that is a little bit challenging because the mountain because of the shape of the mountain and and all those complications are involved over there but we can simplify that by assuming assuming our mountain to be made of a cylinder now I know I know what you're gonna say you're gonna say that's ridiculous but that's what we do in physics we make such ridiculous assumptions I mean we make assumptions like this to make the math easier and we would say well ok the final answer may not be accurate but we sort of get an order of merit estimate we can approximate you know some ballpark figure all right so if we assume or mount our mountain to be a perfect cylinder and let's say the cylinder has a height Hetch high attached now can we calculate what will be the pressure at the bottom well yes now we can do that the pressure at the bottom we would be the force that this whole mountain is putting the cylinder is putting in the bottom the force and that would be the weight of that cylinder cylindrical mountain divided by the area the area that we're considering now since we don't know what the mass of the stuff that's made of over here we could write mass as density times volume of the cylinder density times volume but the volume is just high times the area and so the pressure pretty much I shouldn't write equal to I should put here approximation these are approximations approximately Rho G H so if this pressure exceeds this number then there's a good chance our rocks will start flowing in the mountain will sink and so that's how we can figure out what's the maximum value of Hetch so we'll just say the pressure at the bottom Rho gh should be less than less than the elastic limit elastic limit of the materials over here all right we know what G is that's the acceleration due to gravity height is something we need to calculate density another thing that we don't know again density depends on which material we're dealing with but we could say okay this is solid density of water is thousand and solids are usually denser than water and so we could say okay the entity must be more than thousand some solids have density 2000 3000 some can even have 5000 a good way to pick a number over here is you look on the right hand side you say look there's a three over there so just to cancel things out and keep our number simple we'll assume our density to be 3000 all right yeah so we're gonna take density to be I don't think people do it this way but anyways we could say pretty much density is about three thousand kilograms per meter cube I'm not gonna put the limit the unit's over here this 3000 G is 10 let me use a different color for this 10 and the height that we want well that's hatch then there should be less than 3 times 10 to the power 8 8 meters so if we cancel things the 3 cancels that's the reason we chose that and you have 4 zeros and your 4 zeros over here cancel and so H turns out to be less than about 10 to the power 4 meters or H becomes less than 10 kilometers now we know that this is not a very accurate answer because of all the approximations that we did but a good way to see how close we are is by looking at a list of all the tallest mountains of the world and so if we go to Wikipedia it's showing me now the tallest mountains of the world and their heights over here Mount Everest has about 8.8 this is in meters this is in feet we look at meters about 8.8 kilometers and if you look at all the other tall mountains then you will see that their heights are pretty much between eight to nine kilometers I don't know about you but I'm feeling very good very good about our rough calculations it's unbelievable that we got this number but there was one mountain which was not listed in that and that is Mount Olympus Mount Olympus has a height of so let me just write it down Mount Olympus Oh impass Mount Olympus has a height of 25 kilometers and you were like what way taller than terrorist yes it's not listed because it's on Mars alright so this thing is on Mars but here's the big question why is the mountain on Mars so much taller than whatever estimate we did over here I mean we saw this is a pretty good estimate and the physics should be pretty much the same whether you're doing it on earth or Mars right so where do you think what do you think would be different in these cat in this calculation if we were doing it on Mars think about it