APFIVE Gravitational Field with Variable Density
A planet has a density that varies with distance $$r$$ from its center as $$\rho(r) = \rho_0 \left(1 - \frac{r^2}{R^2}\right)$$, where $$R$$ is the planet’s radius and $$\rho_0$$ is a constant. What is the gravitational field strength at a distance $$r < R$$ from the center?
A
$$(4\pi G \rho_0/3)r$$
B
$$G\rho_0(1 - r^2/R^2)r$$
C
$$\frac{4\pi G \rho_0}{3}\left(r - \frac{r^3}{5R^2}\right)$$
D
$$(4\pi G \rho_0/3)(r - r^3/R^2)$$
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