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Usually when calculating the potential energy of a body it is sufficient to take its center of gravity’s distance from the ground in order to get a result according to the formula $E_p=g*h*M$. But the overall potential energy of a body should be better described as the sum of the potential energy of each and every infinitely small component of said structure, which (for a body having homogeneous density) would vary solely depend on the component’s distance from the ground.

As such, I believe that a better approach to computing the total potential energy of a body - especially if it is required great precision or if the body has “important” dimensions - would be to define it as an integral. Here is where I have to ask an input on how such an integral should be constructed: suppose I have one such very large object of conical shape with the base found at height h from the ground and the tip at a higher height H. How should an integral be written to calculate the incognita, if we assume the density and gravitational force are constant?

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  • $\begingroup$ I don’t see why integrating the contributions of the particles to gravitational potential energy would give you a result more precise than simply using the center of mass if you are assuming constant gravitational force. $\endgroup$
    – Bob D
    Commented Dec 1, 2023 at 10:00
  • $\begingroup$ I understand. What if the gravitational force was not constant instead, hypothetically? Would an integral be useful then? $\endgroup$
    – Riccardo Zanardi
    Commented Dec 1, 2023 at 10:14
  • $\begingroup$ It would certainly be more interesting. But it would depend on the desired level of precision and how “tall” the object is $\endgroup$
    – Bob D
    Commented Dec 1, 2023 at 10:42
  • $\begingroup$ I see. Thanks for the input $\endgroup$
    – Riccardo Zanardi
    Commented Dec 1, 2023 at 11:16

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If you take a horizontal slice of the cone of at height $x$ width depth $\delta x$ then this has volume $\pi r(x)^2 \delta x$ where $r(x)$ is the radius of the cone at height $x$ (assuming a solid cone). If the cone is uniform with density $\rho$ then the mass of the slice is $\delta m = \rho \pi r(x)^2 \delta x$ and its potential energy is

$xg\delta m = \rho \pi g x r(x)^2 \delta x$

and so the total potential energy of the cone is

$g \int_h^H \rho \pi x r(x)^2 \space dx$

But $\int \rho \pi x r(x)^2 dx$ is exactly the integral that you would do to find the location of the centre of mass of the cone, which turns out to be one quarter of the distance from the base of the cone to its vertex. So you can find the cone's potential energy by assuming its whole mass is at its centre of mass.

In general, you can calculate the potential energy of any object by assuming its whole mass is concentrated at its centre of mass as long as the acceleration due to gravity, $g$, is uniform across the whole object.

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  • $\begingroup$ I see. So I would guess then that an integral is better used when is the gravitational acceleration that’s varying significantly while acting on a body? That’s beside the original question of course, I’m asking now out of curiosity. $\endgroup$
    – Riccardo Zanardi
    Commented Dec 1, 2023 at 10:13
  • $\begingroup$ Yes, if the object is so large that gravity varies significantly from one part of it to another then you have to go back to first principles and use the integral method. $\endgroup$
    – gandalf61
    Commented Dec 1, 2023 at 10:21
  • $\begingroup$ I see. Thanks for the answer $\endgroup$
    – Riccardo Zanardi
    Commented Dec 1, 2023 at 10:35

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