Newton's Law of Gravitation — Question 2

Question 2

Equal gravitational forces can produce very unequal accelerations

Two isolated point masses $M=9m$ and $m$ are separated by $r$. Work in an inertial frame and neglect all other interactions.

1. Compare force magnitudes and acceleration magnitudes.

2. Find the acceleration of their center of mass and explain the distinction.

Question 2 - Solution

Equal gravitational forces can produce very unequal accelerations

(a) Derivation and calculation

Take $+x$ from $M$ toward $m$. Newton’s third law gives $$\vec F_M=+\frac{9Gm^2}{r^2}\hat x,\qquad \vec F_m=-\frac{9Gm^2}{r^2}\hat x.$$ Dividing each force by its own mass, $$a_M=\frac{Gm}{r^2},\quad a_m=\frac{9Gm}{r^2},\quad \boxed{a_m/a_M=9}.$$

(b) Result and interpretation

For the isolated pair, $$\vec a_{\mathrm{CM}}=\frac{M\vec a_M+m\vec a_m}{M+m} =\frac{\vec F_M+\vec F_m}{10m}=\boxed{\vec 0}.$$ Equal interaction forces give unequal individual accelerations while preserving the center-of-mass velocity.

Physical check

The larger mass accelerates less; the force pair cannot accelerate the isolated mass center.