Newton's Law of Gravitation — Question 3

Question 3

A force measurement determines a mass product rather than each mass

Two positive point masses have known sum $S$. A force measurement at known separation $r$ gives $Fr^2/G=3S^2/16$.

1. Find the two masses, allowing interchange of their labels.

2. What upper bound on $Fr^2/G$ must any pair of positive masses with sum $S$ obey?

Question 3 - Solution

A force measurement determines a mass product rather than each mass

(a) Derivation and calculation

Let $x=m_1/S$, so $m_2=S(1-x)$: $$x(1-x)=\frac3{16},\quad x^2-x+\frac3{16}=0.$$ $$x=\frac{1\pm\sqrt{1-3/4}}2=\frac14,\frac34, \qquad \boxed{\{m_1,m_2\}=\{S/4,3S/4\}}.$$

(b) Result and interpretation

Completing the square yields the admissibility bound $$m_1m_2=m_1(S-m_1)=\frac{S^2}{4}-\left(m_1-\frac S2\right)^2.$$ Thus $$\boxed{0<\frac{Fr^2}{G}\le\frac{S^2}{4}},$$ with equality only for $m_1=m_2=S/2$. A larger inferred product is incompatible with the stated model.

Physical check

The two roots exchange mass labels and reproduce both the specified sum and product.