# Does this proof of "all bodies fall with the same acceleration" presuppose the identity of gravitational mass and of inertial mass?

It seems to me that one way to show that all material objects have the same acceleration in the gravitational field of the Earth (independently of their mass) one can make use of these two formulas.

I use $$m_A$$ for the mass of object A and $$m_T$$ for the mass of the Earth, $$d$$ for the distance from the Earth's center, and $$a_A$$ of A's acceleration.

One formula gives the value of the gravitational force $$F = G\frac{m_A m_T}{d^2}\tag1$$

The other (Newton's second law) expresses force as: $$F=m_A a_A\tag2$$

Equations (1) and (2) imply that $$G\frac{m_A m_T}{d^2}=m_A a_A\tag3$$

and finally that

$$a_A= \frac {G m_A m_T/d^2}{m_A}=G\frac{m_T}{d^2}.\tag4$$

From this one sees that acceleration does not contain any factor expressing the mass of object A, which means that acceleration is the same for all objects A (whatever their mass may be).

Out of this arises my question: I could not have written equation (4) if I had not considered $$m_A$$ to be the same quantity in the numerator and in the denominator; so it seems that the proof presupposes the identity of gravitational mass and of inertial mass; in that case, how comes it is said that, from the point of view of newtonian mechanics, these two masses are definitionnaly different (though they happen - contingently - to have the same quantitative value) a feature that distinguishes newtonian mechanics from einsteinian relativistic one?

It seems to me that the fact that all bodies fall with the same acceleration (under the Earth's attraction) is a fundamental result of newtonian mechanics. But how can this be the case if inertial mass and gravitational mass are not considered a priori as identical?

Note: the claim that the two kinds of masses are not identified a priori in newtonian mechanics is taken from Einstein's Evolution of physics.

• @Qmechanics.- Thanks for your edit. Aug 11, 2021 at 22:42

Your question touches upon a fundamental aspect of what distinguishes Newtonian Gravity from Einstein Gravity.

As you correctly point out, when gravity was first being investigated, it was found experimentally that all objects fall with the same acceleration. This is consistent with assuming that gravitational mass and inertial mass are equivalent, as you showed in your proof. But this is only an assumption that comes after an experiment - there is no fundamental principle that says the two must be equal. In Newton's theory of gravity, there is no way to prove, from first principles, that they must be equal.

Einstein Gravity is different. To summarise General Relativity, gravity is not a force. Instead, what we perceive as the force of gravity is instead a particle moving along a straight line (a "geodesic") in a curved spacetime. General Relativity never has to make reference to mass, and because of that, we can immediately say that all objects must follow the same path (and so show the same gravitational behaviour) regardless of their mass.

So Newtonian Gravity can't prove the two masses (gravitational and inertial) are the same from first principles, whereas Einstein Gravity can, because of their different approaches to describing gravity.

Your proof does not prove that inertial mass and gravitational mass are identical- it shows that they are proportional to each other. If, say, inertial mass was twice gravitational mass, your proof would still hold but with a halved value for G.

• I understand the difference, though true, I hadn't noticed that. So I should have asked " does this proof pressupose their proportionality?" Aug 11, 2021 at 17:07
• It is not an assumption- it is a relation proved by observation. Aug 11, 2021 at 19:15

The fact that all bodies fall with the same acceleration comes from experience. Also by observation we know the movement (relation acceleration x distance) of the planets around the Sun. So the expression:$$a = \frac{GM}{d^2}$$ is more fundamental than the expression for the gravitational force (M being the big mass around which much smaller bodies move).

On the other hand, if we associate the force to a spring deflection, we can relate force and acceleration for a given body. For example rotating it with a constant angular velocity. We see a proportionality between force (spring deflection) and acceleration (centripetal in this example). Besides that, more massive bodies shows greater acceleration for the same force. So, we have the Newton's second law, and the inertial mass as the proportionality: $$m_i = \frac{F}{a}$$

Finally, we can simply hang those bodies from the spring and record its deflection. In order to fulfill Newton's second law, we postulate that there is a downward force of gravity, otherwise there would be only an upward force from the spring on the object in spite of it shows no acceleration. More massive bodies result in bigger deflection of the spring. We can observe here a proportionality: $$F = Cm_i$$ where $$C$$ is a constant.

The point is that the last result comes only from experience. In principle we could imagine that $$F$$ would be proportional to some function of $$m_i$$ so that what we could call gravitational mass would be $$m_g = f(m_i)$$, and not necessarily linear.

In Newtonian mechanics there is no theoretical reason to justify that $$m_i = m_g$$, in the sense explained above. Not even that they are linearly proportional.