Answer to Question #170765 in Classical Mechanics for Hridita Chowdhury

Question

Compute the gravitational potential & gravitational field due to a hollow sphere :

1. At a point outside a sphere

2.At a point inside the hollow space

3.At a point inside the material

Answer 1

1)

$F=\frac{3GMm}{r^2R^3}\int _0^R x^2dx=\frac{GMm}{r^2},$

2)

$F=\frac{GMm}{R^2}\cdot \frac rR,$

3)

$F=\frac{4\pi Gm}{r^2}\int _{R_a}^r R^2 \rho dR=\frac{GMm}{r^2}\int_{R_a}^{r}\frac{3R^2}{R_b^3-R_a^3}dR=\frac{GMm}{r^2}\cdot \frac{r^3-R_a^3}{R_b^3-R_a^3}.$