Newton's Third Law — Question 7

Question 7

Earth responds to the same mutual gravitational force

A 2 kg object and an idealized Earth of mass $6\times10^{24}$ kg exert equal gravitational forces of 20 N on one another. Ignore all other forces for this comparison.

1. Find each acceleration magnitude.

2. Find their ratio.

3. Explain why treating Earth as nearly stationary does not deny the third law.

Question 7 - Solution

Earth responds to the same mutual gravitational force

(a) Apply the same force to different masses.

$$a_o=20/2=10,\quad\boxed{a_E=\frac{20}{6\times10^{24}}=3.333\times10^{-24}}\quad\mathrm{m/s^2}$$ The object has acceleration $20/2=10$ m/s$^2$. Earth has acceleration $20/(6\times10^{24})\approx3.333\times10^{-24}$ m/s$^2$, toward the object.

(b) Compare responses.

$$\frac{a_o}{a_E}=\frac{M_E}{m_o}=\frac{6\times10^{24}}2=3\times10^{24}$$ The object-to-Earth acceleration ratio is $3\times10^{24}$, equal to the inverse mass ratio.

(c) State the approximation.

$$F_{o\to E}=F_{E\to o}=20\,\mathrm N\quad\text{(magnitudes)},\qquad a_E\ll a_o$$ Earth’s enormous inertia makes its response negligible at ordinary precision. Approximating its motion as fixed concerns the size of its acceleration, not the existence or magnitude of the force on it. These are supplied idealized values, not a local-field measurement.