Gravitation and the Principle of Superposition — Question 9

Question 9

Two field measurements can identify two fixed source masses

Point sources $M_1$ and $M_2$ lie at $x=0$ and $x=3a$. At $x=a$, the signed field is zero. At $x=-a$, its rightward magnitude is $5GM_0/(4a^2)$, where $M_0$ is a reference mass.

1. Infer $M_1$ and $M_2$.

2. Check the result against both signed field equations.

Question 9 - Solution

Two field measurements can identify two fixed source masses

(a) Derivation and calculation

Normalize fields by $G/a^2$. The two measurements give $$-M_1+\frac{M_2}{4}=0,\qquad M_1+\frac{M_2}{16}=\frac54M_0.$$ From the first equation $M_2=4M_1$, hence $$\frac54M_1=\frac54M_0,\qquad \boxed{M_1=M_0,\quad M_2=4M_0}.$$

(b) Result and interpretation

Substitution into the original signed equations gives $$g(a)=\frac{GM_0}{a^2}\left(-1+\frac44\right)=0,$$ $$g(-a)=\frac{GM_0}{a^2}\left(1+\frac4{16}\right) =\frac{5GM_0}{4a^2}>0.$$ The measurement equations are independent, so both source masses are determined.

Physical check

Both inferred masses are positive and both measured directions are reproduced.