Gravitation

Question

Two identical masses $m$ are a distance $l$ apart. The masses interact only via gravitation. Starting from rest, how long does it take before they collide? Assume $l \gg r$ where $r$ is the radius of the masses.

Initial Energy

$U_0=-\frac{Gm^2}{l}$

Conservation of Energy

$r$ is the distance between the two masses.

$\frac{1}{2}mv^2(r)-\frac{Gm^2}{r}=-\frac{Gm^2}{l}$

$v(r)=-\sqrt{2Gm\left(\frac{1}{r}-\frac{1}{l}\right)}$

$v=\frac{dr}{dt}$

$\frac{dr}{dt}=-\sqrt{2Gm\left(\frac{1}{r}-\frac{1}{l}\right)}$

$\int_{l}^{0}\sqrt{\frac{lr}{l-r}}\,dr=-\int_{0}^{T}\sqrt{2Gm}\,dt$

Integration Table:

$\int\sqrt{\frac{r}{l-r}}\,dr=-\sqrt{r(l-r)}-l\arctan\frac{\sqrt{r(l-r)}}{r-l}$

$\int_{l}^{0}\sqrt{\frac{r}{l-r}}\,dr=\sqrt{l(l-l)}+l\arctan\frac{\sqrt{r(l-l)}}{l-l}$

We have problems so let’s take the limit of an increment going to zero.

$\int_{l-\delta}^{0}\sqrt{\frac{r}{l-r}}\,dr=\sqrt{\delta(l-\delta)}+l\arctan\frac{\sqrt{\delta(l-\delta)}}{-\delta}$

$\lim_{\delta\to0}\sqrt{\delta(l-\delta)}+l\arctan\frac{\sqrt{\delta(l-\delta)}}{-\delta}=-l\lim_{\delta\to0}\arctan\frac{\sqrt{\delta(l-\delta)}}{\delta}$

$-l\lim_{\delta\to0}\arctan\frac{\sqrt{\delta(l-\delta)}}{\delta}=-l\lim_{\delta\to0}\arctan\sqrt{\frac{l-\delta}{\delta}}=-\frac{l\pi}{2}$

$-\frac{l\sqrt{l}\pi}{2}=-\sqrt{2Gm}T$

$T=\frac{\pi l\sqrt{l}}{2\sqrt{2Gm}}$

$T=\frac{\pi l}{2}\sqrt{\frac{l}{2Gm}}$

2013 F=ma Exam Problem 16
A very large number of small particles forms a spherical cloud. Initially they are at rest, have uniform mass density per unit volume $p_0$ , and occupy a region of radius $r_0$ . The cloud collapses due to gravitation; the particles do not interact with each other in any other way. How much time passes until the cloud collapses full?