Newton's Law of Gravitation — Question 1

Question 1

A surface gap is not the distance in the inverse-square law

Two nonoverlapping uniform spheres have radii $R$ and $2R$, masses $M$ and $4M$, and surface gap $R$. For this problem use the spherical-body result that their attraction equals that of masses at their centers.

1. Find their force in units of $GM^2/R^2$.

2. If the surface gap doubles, find the ratio of the new force to the old force.

Question 1 - Solution

A surface gap is not the distance in the inverse-square law

(a) Derivation and calculation

Exterior spherical symmetry permits center masses. For a gap $s$, $$r=R+s+2R=3R+s,\qquad F(s)=\frac{4GM^2}{(3R+s)^2}.$$ At $s=R$, $$\boxed{F_1=\frac{GM^2}{4R^2}},\qquad \frac{F_1}{GM^2/R^2}=\frac14.$$

(b) Result and interpretation

Doubling the gap gives $$s_2=2R,\quad r_2=5R,\quad F_2=\frac{4GM^2}{25R^2}.$$ Therefore $$\boxed{\frac{F_2}{F_1}=\left(\frac{4R}{5R}\right)^2=\frac{16}{25}}.$$

Physical check

The center separation increases by $R$, so doubling the gap does not halve the gravitational force.