Newton's Law of Gravitation — Question 5

Question 5

Force ratios test the exponent without knowing the calibration constant

An experiment is modeled by $F=Cr^{-n}$, with fixed source masses and an unknown positive constant $C$. At separations $r$ and $3r$, the measured force ratio is $F(r)/F(3r)=9$.

1. Infer $n$ and predict $F(2r)/F(r)$.

2. Explain why a common multiplicative calibration error cancels but an additive force offset generally does not.

Question 5 - Solution

Force ratios test the exponent without knowing the calibration constant

(a) Derivation and calculation

For fixed $C$, $$\frac{F(r)}{F(3r)}=\frac{Cr^{-n}}{C(3r)^{-n}}=3^n=9.$$ $$\boxed{n=\frac{\ln9}{\ln3}=2},\qquad \boxed{\frac{F(2r)}{F(r)}=2^{-2}=\frac14}.$$

(b) Result and interpretation

A common gain $k$ cancels exactly: $$\frac{kF_1}{kF_2}=\frac{F_1}{F_2}.$$ For an additive offset $b$, $$\frac{F_1+b}{F_2+b}-\frac{F_1}{F_2} =\frac{b(F_2-F_1)}{F_2(F_2+b)}.$$ The error is generally nonzero, provided the measured denominator is nonzero. Additional separations can test consistency.

Physical check

The exponent is dimensionless and the predicted force decreases with distance.