I am trying to calculate the gravitational potential of a solid block, and I have a nonlinear answer which strikes me as wrong.

A block with horizontal surface area $A [m^{-2}]$ and uniform density $\rho[kg\cdot m^{-3}]$ has its base at height $H_0[m]$ and its top at height $H_1$, total mass is $m=\rho A(H_1-H_0) [kg]$.

Considering horizontal slices in a uniform gravity field, I wrote down this for total GPE:

$\int_{H_0}^{H_1}\mathrm{d}E=\int_{H_0}^{H_1}gh\ \mathrm{d}m=\int_{H_0}^{H_1}\rho Agh\ \mathrm{d}h=\rho Ag\int_{H_0}^{H_1}h\ \mathrm{d}h=\frac{1}{2}\rho Ag(H_1^2-H_0^2)$

But this isn't translation-independent, I get a different answer if I change the heights to $H_0+x$ and $H_1+x$. My intuition tells me this is wrong, shouldn't I get the same GPE as for a point mass at height $\frac{H_0+H_1}{2}$, which would be translation-independent?

What is my mistake?


1 Answer 1


Your answer is $$ \frac 12 \rho g A (H_1^2-H_0^2)= \rho g A (H_1-H_0)[(H_1+H_2)/2]\\ = \hbox{density$\times$ volume} \times g (H_1+H_2)/2\\ = \hbox{Mass} \times g \times \hbox{height of center of mass}\\ = Mgh $$ which is correct.


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