- #1
burakumin
- 84
- 7
The gravitational potential energy of two massic points ##P_1## and ##P_2## with respective masses ##m_1## and ##m_2## is given by
$$U = -G \frac{m_1 m_2}{|| P_2 - P_1 ||}$$
Now I was wondering how this formula could be applied to continuous matter. Let us imagine a very simple case where we have a segment-like body (a cylinder with negligible radius) of length ##L## and constant linear density ##\rho##. The gravitational potential energy of two infinitesimal segments centered in ##x_1## and ##x_2## (with ##0 \leq x_1 < x_2 \leq R##) would be:
$$\delta U = -G \rho^2 \frac{\textrm dx_1 \textrm dx_2}{x_2 - x_1}$$
But now if we try to add every contribution:
$$U = -G \rho^2 \int_{x_1 = 0}^{x_2} \int_{x_2=0}^L \frac{\textrm dx_1 \textrm dx_2}{x_2 - x_1}$$
this integral diverges...
Of course real matter is not really continuous so a more relevant description of reality would rather be a finite sum of many close but not superposed massic points. However it seems quite unsatisfying that the standard formula of gravitational potential energy does not work with the very common assumption of continuous matter. Am I missing something ?
$$U = -G \frac{m_1 m_2}{|| P_2 - P_1 ||}$$
Now I was wondering how this formula could be applied to continuous matter. Let us imagine a very simple case where we have a segment-like body (a cylinder with negligible radius) of length ##L## and constant linear density ##\rho##. The gravitational potential energy of two infinitesimal segments centered in ##x_1## and ##x_2## (with ##0 \leq x_1 < x_2 \leq R##) would be:
$$\delta U = -G \rho^2 \frac{\textrm dx_1 \textrm dx_2}{x_2 - x_1}$$
But now if we try to add every contribution:
$$U = -G \rho^2 \int_{x_1 = 0}^{x_2} \int_{x_2=0}^L \frac{\textrm dx_1 \textrm dx_2}{x_2 - x_1}$$
this integral diverges...
Of course real matter is not really continuous so a more relevant description of reality would rather be a finite sum of many close but not superposed massic points. However it seems quite unsatisfying that the standard formula of gravitational potential energy does not work with the very common assumption of continuous matter. Am I missing something ?