Find the force of attraction between a homogeneous sphere

Find the force of attraction between a homogeneous sphere of density $\mu=1$, radius $R$, and a material point of mass $m$ located at distance $b$ from the center of the sphere, where $b>R$.

Step 1: Mass of the sphere

\[M=\mu\cdot\frac{4}{3}\pi R^3=\frac{4\pi R^3}{3}\qquad(\mu=1).\]

Step 2: Apply the shell theorem

Since every point of the sphere lies at radius $\le R \lt b$ from the center, and each thin shell attracts the external point $m$ as though its mass were concentrated at the center, the whole sphere does too:

\[F=G\frac{Mm}{b^2}=\frac{4\pi GmR^3}{3b^2}.\]

\[\boxed{F=\frac{4\pi GmR^3}{3b^2}}\]

Step 3: Verify by direct shell integration

A shell of radius $r\le R$ and thickness $dr$ has mass $dM=4\pi r^2\,dr$, and (by the shell theorem for a single shell) contributes $dF=G\dfrac{m\,dM}{b^2}$ to the force on the external point. Summing over all shells from $r=0$ to $R$:

\[F=\frac{4\pi Gm}{b^2}\int_0^R r^2\,dr=\frac{4\pi Gm}{b^2}\cdot\frac{R^3}{3}=\frac{4\pi GmR^3}{3b^2},\]

exactly matching Step 2.

Step 4: Special cases

At the surface ($b=R$): $F=\dfrac{4\pi GmR}{3}$. Far away ($b\gg R$): $F\to0$ like $1/b^2$, as expected for a distant mass.