A spherical cloud of mass $M$ has radius $r(t)$ and initial radius $r(0)=R$. It contains material with uniform mass density $\rho(t)$, and zero pressure. Ignoring the cosmological constant, show that if it is initially at rest at $t=0$ and the subsequent gravitational collapse is governed by Newton's law $\ddot{r}=-G M / r^{2}$, then

$$\dot{r}^{2}=2 G M\left(\frac{1}{r}-\frac{1}{R}\right) .$$

Suppose $r$ is given parametrically by

$$r=R \cos ^{2} \theta$$

where $\theta=0$ at $t=0$. Derive a relation between $\theta$ and $t$ and hence show that the cloud collapses to radius $r=0$ at

$$t=\sqrt{\frac{3 \pi}{32 G \rho_{0}}},$$

where $\rho_{0}$ is the initial mass density of the cloud.