Question

Part 1. (Easy)

1.) Consider a tall block of material with constant cross sectional area. Since the block has mass, it exerts some amount of pressure on the material below it. Ultimately at some height the object's own mass must cause the tower to crumble. What is the height for a material with a density of \rho and an ultimate tensile strength of \sigma_t on a planet with gravity g?

Compute the maximum heights in meters of the following substances:

Iron
Lead
Concrete

Can someone help with this?

Answer 1

This is more of a buckling problem, so I'm confused as to why they gave UTS as a property. For buckling, we need Young's modulus and moment of inertia. We lack both in this case. (We can pull Young's modulus from UTS but we need the strain, which isn't given either.)

Let's assume the material behaves the same in compression as it does in tension.


For the case of a material that has the same properties in compression as in tension, we need to find the stress at the bottom of the column. This is given as\[\sigma_B = {F \over A}\]
We need to express F in terms of density and gravity. \[F = \int\limits_0^L \rho A g dy\]where A is the cross sectional area and dy is an infinitesimally small height of the column.

Solving the integrand \[F = \rho A g L\]

Substituting into our equation for stress\[\sigma = {\rho A g L \over A} = \rho g L \]

We can find the maximum length as \[{\sigma_t \over \rho g} = L\]

Answer 2

Looks right. Thanks!