A2
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Calculate the gravitational potential energy $E_G$ of the NU at a time $t$ when its radius is equal to $R$. Give your answer as a formula expressing $E_G$ in terms of $M$ and $R$.
Remark: if the mass of the system increases from $m$ to $m + \Delta m$ without any changing the relative mass distribution, the gravitational potential energy increases by
\[ \Delta E_G = \int\limits_0^{\Delta m} \varphi \, dm,\]
where $\varphi$ is a gravitational potential at the point of mass $dm$. Also $\varphi$ is chosen so that $\varphi=0$ on infinity.
H1
Use Guass's law for the gravitational field
\[\int_\Gamma \vec{g} \cdot d\vec{S} = -4\pi Gm,\] where $m$ is the mass enclosed within the closed surface $\Gamma$ to find a gravitational field and a gravitational potential in the universe.
H2
The gravitational field $g(r)$ in the massive ball with radius $R$ and mass $M$
\[ g(r) = GM \frac{r}{R^3}\]
H3
The gravitational potential $\varphi(r)$ in the massive ball with radius $R$ and mass $M$
\[ \varphi(r) = -\frac{3GM}{2R} + \frac{GMr^2}{2R^3}.\]
