Gravitation and the Principle of Superposition — Question 1

Question 1

The field cancellation point lies nearer the smaller source

Fixed point sources $M$ and $9M$ lie at $x=0$ and $x=d$. A negligible test mass can be placed on their line.

1. Find every finite point on the line where the total gravitational field is zero.

2. Explain why adding field magnitudes instead of signed components would miss the cancellation.

Question 1 - Solution

The field cancellation point lies nearer the smaller source

(a) Derivation and calculation

For $0<x<d$, the left source attracts left and the right source attracts right: $$g_x=-\frac{GM}{x^2}+\frac{9GM}{(d-x)^2}.$$ Setting $g_x=0$ and taking positive distances, $$d-x=3x,\qquad \boxed{x=d/4}.$$ For $x<0$ both fields point right; for $x>d$ both point left. There are no other finite zeros.

(b) Result and interpretation

At the cancellation point, $$g_{1x}=-\frac{16GM}{d^2},\quad g_{2x}=\frac{9GM}{(3d/4)^2}=\frac{16GM}{d^2}.$$ $$g_{1x}+g_{2x}=0,\qquad |g_{1x}|+|g_{2x}|=\frac{32GM}{d^2}\ne0.$$ Cancellation is a signed vector sum, not a sum of magnitudes.

Physical check

The zero lies nearer the smaller source, where its weaker mass is compensated by a shorter distance.