Compression of an object under it's own weight.

[LordGfcd]

Let's first consider a cube side length a, mass m, Young's modulus of the block is E. How do we calculate the decrease of the height of the center of mass of that cube ?

 

[Chestermiller]

What are your thoughts so far on how to approach this problem?

 

[LordGfcd]

Actually I solved this problem assuming the cube is deform before the normal force make balance with gravity. I eventually find an acceptable result (a=10cm,m=1kg,E=10^7 Pa) : 2,27 μm. But I still think my approaching is wrong because I didn't consider the normal force (action equal minus reaction ofcourse). So I must consider the normal force too, but I don't know how . And, if the normal force balance with gravity, isn't the cube will stop deforming ?

 

[Chestermiller]

The differential force balance on the section of the cube between z and z + $\Delta z$ (z is measured downward from the top) is $$a^2\frac{d\sigma}{dz}=\rho g a^2$$ where A is the cross sectional area, $\sigma$ is the compressive stress, and $\rho$ is the density of the material $m/(a^2L)$. Do you see how this result is derived?

 

[LordGfcd]

Thank you very much, I was completely wrong with my argument.