Using Newtonian field equation to find the gravity inside a sphere

[lostminty]

Homework Statement

work out g field at a distance r=R/2 from the centre of a spherically symmetric body of radius R.

Homework Equations

$\nabla^{2}\psi=4\pi G \rho$

$\psi=G\int_{V'}\rho\frac{1}{|\vec{r}-\vec{r^{'}}|}d^{3}\vec{r^{'}}$

$-\int_{V}\nabla\cdot g dV = \int_{V}\rho dV$

V=4/3∏r³

The Attempt at a Solution

V=4/3∏r³

V₁=4/3∏(R/2)³ = 1/6∏R³
V₂=4/3∏R³
$\int_{1/6\pi R^{3}}^{4/3\pi R^{3}}\rho dV$

$=\rho(4/3\pi R^{3}-1/6\pi R^{3})$

$=\frac{7}{6}\rho \pi r^{3}$

 

[lostminty]

Homework Statement

work out g field at a distance r=R/2 from the centre of a spherically symmetric body of radius R.

Homework Equations

$\nabla^{2}\psi=4\pi G \rho$

$\psi=G\int_{V'}\rho\frac{1}{|\vec{r}-\vec{r^{'}}|}d^{3}\vec{r^{'}}$

$-\int_{V}\nabla\cdot g dV = \int_{V}\rho dV$

V=4/3∏r³

The Attempt at a Solution

V=4/3∏r³

V₁=4/3∏(R/2)³ = 1/6∏R³
V₂=4/3∏R³
$\int_{1/6\pi R^{3}}^{4/3\pi R^{3}}\rho dV$

$=\rho(4/3\pi R^{3}-1/6\pi R^{3})$

$=\frac{7}{6}\rho \pi r^{3}$

should be a G*2*pi in the front of one of those integrals