Does the acceleration of Earth depend on its mass?

Question

Let's suppose that an apple is falling down towards the Earth.

• From Newton's law of universal gravitation: the force exerted on Earth by the apple is $$F = \frac{GM_1 M_2 }{r^2}$$

• From Newton's second law: the force exerted on Earth by the apple is $$F =\underbrace{M_1}_{\rm Earth's\ mass} \times \underbrace{a}_{\rm Earth's\ acceleration}$$

So $$M_1 × a = \frac{G M_1 M_2} {r^2} $$

The result is that $$ a = \frac{GM_2}{r^2}$$

This result is a dilemma for me because :

1. It indicates that acceleration of Earth due to gravity doesn't depend on Earth’s mass!

2. On the other hand, a lot of references say that acceleration is inversely proportional to mass, therefore, The acceleration of earth depends on its mass!

I hope someone helps me overcome this dilemma.

Answer 1

For a given force acting on an object, the acceleration of such object will be inversely proportional to its mass. But in the case of gravity, the force is also dependent on the mass, so if you change the mass you also change the force, in such a way that the acceleration of the object due to another mass is independent of its own mass. There is no contradiction with the first statement because the force is no longer the same, you are changing it when you change you object's mass.

Answer 2

The force exerted on the apple by the Earth is equal to the force exerted on the earth by the apple.
Because F=ma, and because the apple's mass is much less than the Earth's mass, the apple's acceleration is much greater than the Earth's acceleration ( but of course in the opposite direction).

Your mistake can be avoided if you ignore force and only calculate accelerations. Acceleration of the apple is proportional to the Earth's mass; acceleration of the Earth is proportional to the apple's mass. After all, gravity accelerates free masses; it exerts forces on masses only indirectly.

Answer 3

It does depend on the earth's mass and because of this dependence that you got such a small acceleration of the earth in spite of a sufficient force acting on the earth.

And acceleration is still inversely proportional to mass since

$$a = \frac{F}{m}$$

And what you did above is that just canceled the mass in the $F$ term with the one in the denominator.

So the earth does accelerate but you can calculate its value, it's very small in comparison to the acceleration of the apple because

$$a_{earth} = \frac {GM_{apple}}{r^2}$$ and

$$a_{apple} = \frac{GM_{earth}}{r^2}.$$

Here the masses of the earth and the apple makes the difference.