Newton's Law of Gravitation — Question 8

Question 8

Geometrically similar spheres do not keep the same contact force

Two equal uniform spheres of density $\rho$ and radius $R$ just touch. Use $M=4\pi\rho R^3/3$ and the exterior spherical-body force law.

1. Derive how their contact attraction scales with radius.

2. If every radius doubles at fixed density, compare force and acceleration of either sphere.

Question 8 - Solution

Geometrically similar spheres do not keep the same contact force

(a) Derivation and calculation

At contact the center separation is $2R$: $$M=\frac{4\pi}3\rho R^3,\qquad F=\frac{GM^2}{(2R)^2} =\boxed{\frac{4\pi^2}{9}G\rho^2R^4}.$$ Thus $F\propto R^4$ at fixed density.

(b) Result and interpretation

Under $R\mapsto2R$, $$\frac{M'}M=2^3=8,\quad \boxed{F'/F=2^4=16}.$$ For either sphere, $$a=\frac FM=\boxed{\frac\pi3G\rho R},\qquad \boxed{a'/a=16/8=2}.$$

Physical check

The dimensions of $G\rho R$ are acceleration; changing radius also changes both masses.