# Estimating the force needed to increase height of a mountain?

How would you estimate the force required by a tectonic plate to make the height of a mountain increase when it pushes against another? I've used a method to try and do it for Mt Everest and have ended up with 8x10^(-7)N required to increase its height which I don't think sounds reasonable

By the way, I was doing this for fun (it doesn't have any real life value, I don't think)

• Use the Force of gravity, Luke! F=m*g. m is the mass of your mountain and g is Earth's gravitational surface acceleration. Commented Sep 28, 2014 at 19:31
• After a point the height of a mountain cannot increase anymore lest it will collapse on its own weight. This is somehow analogous to the fact that scale-invariance is no that invariant (or in other words there are many concepts of scale involved and scale-invariance does not apply to all). There is an argument that a horse scaled down by e.g 0.5 will NOT have the strength scaled by same (or analogous) amount and so on Commented Sep 30, 2014 at 2:41

When tectonic plates collide, the crust can become thicker at the edge of collision by the folding and faulting of crustal rocks. Because crust has a lower density than the asthenosphere and mantle, the region of thicker crust can rise due to buoyancy forces until it reaches isostatic equilibrium. This model of orogeny is referred to in geology as isostasy.

Therefore the "force acting to increase the height of a mountain" you wish to estimate is the buoyant force on the mountain/root system. And it will go to zero approaching equilibrium. Use dimensional analysis to write down an equation for static equilibrium:

[mountain weight force]-[buoyant force] = 0

m(h+H)Ag - MHAg = 0

Where m and M are the mass density of the crust and mantle respectively. Where h and H are the mountain's height and the mountain-root's depth respectively. The horizontal cross-section area of the mountain is A. The gravitational acceleration is g.

Think about this and you'll see how it may explain how erosion of mountain-tops has caused the uplift of the Cascades, whereas eruption of seafloor basalts has led to the sinking of the Hawaiian islands.

In really simple terms the equation f=mg can be utilised. It calculates the gravitational potential energy of the mountain and therefore the energy needed to lift a mountain.