Apple and Earth: mutual gravitational forces

Problem

An apple falls from a tree. What is the force with which the Earth attracts the apple, and what is the force with which the apple attracts the Earth? Why don’t we see the Earth move towards the apple?

The two gravitational forces that the Earth and the apple exchange.

Solution

The law of universal gravitation gives the magnitude of the force between the two bodies: $F = G\frac{M_T\, m}{r^2}$ This is a single force, the same for both: by Newton’s third law the Earth attracts the apple with exactly the same magnitude with which the apple attracts the Earth. The two forces are equal and opposite: $F_{T\to m} = F_{m\to T} = G\frac{M_T\, m}{r^2}$ The difference lies in the accelerations, because the masses are enormously different. For the apple: $a_\text{apple} = \frac{F}{m} = \frac{G M_T}{r^2} \approx g \approx 10~\text{m/s}^2$ For the Earth, with $M_T \approx 6\cdot 10^{24}$ kg: $a_\text{Earth} = \frac{F}{M_T} = \frac{G\,m}{r^2} \sim 10^{-25}~\text{m/s}^2$ The Earth shifts by a fraction of an atomic diameter: a motion that is completely invisible. $\ev{F_{T\to m} = F_{m\to T} = G\frac{M_T\,m}{r^2}, \qquad a_\text{Earth}\sim 10^{-25}~\text{m/s}^2}$

Links

Topics: Gravitation Concepts: Law of universal gravitation · Newton’s third law Skills: Symbolic set-up