Why does gravity cause objects to pull other objects with constant acceleration and not constant force?

Question

I know that this may come off as an incredibly dumb question, but please hear me out for a while.

Why don't objects just tug at other bodies with a constant force? Instead, why do they apply different forces depending on the masses of the bodies in consideration, so as to keep the acceleration produced by them fixed for every body? I mean, why not just pull everything with a force of $10$ $N$, for example? How does this intuitively make sense?

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The answers provided in this post all point to the same thing: that acceleration is the same because the effects of force and inertia compensate for each other (as I had tried to point out above). My question, however, starts off from the very point these answers end in, why is acceleration constant?

I have seen a lot of replies and answers about how halving the mass of a body doesn't make it fall faster or slower, but at a constant rate. This does give an intuition about how we can clearly see in our world that acceleration is independent of mass, but this doesn't answer why? Is the fact that these two effects of force and inertia exactly balance out in the world we live in, a miracle? If not, what is the reason behind it?

Are these questions even worth asking?

Answer 1

Imagine that you have an object of mass $M$ attracted to the earth, with the force acting on the object being $F$. You are asking if we can prove intuitively why $F$ should be proportional to $M$ - i.e., $F=gM$ instead of a constant.

A thought experiment attributed (perhaps wrongly?) to Galileo suggest the following: Break the object into two pieces, each with mass $M/2$, hold them together and let them fall together. It makes intuitive sense breaking the object doesn't change its fall at all, so the two halves fall with exactly the same acceleration. Which given Newtons's force = mass times acceleration, the force on each half-mass is half the original $F$.

In other words, we showed that halving M halves F, and in general the same argument shows that multiplying M by any $\alpha$ will result in gravitational force $\alpha F$, in other words $F$ is proportional to $M$, which is what we set out to prove.

Answer 2

In general objects don't experience a constant acceleration due to gravity. But near the surface of the Earth they nearly do.

The gravitational force acting on a pair of objects depends on their masses ($M$ and $m$) and distance between their centers of mass ($r$): $$ F = \frac{GMm}{r^2}.$$

Here $G$ is Newton's gravitational constant. This is Newton's law of universal gravitation.

If we want calculate the acceleration of an object with mass $m$ due to another much bigger object with mass $M$, we can use Newton's second law:

$$\begin{align} F_\mathrm{net} &= m\, a \\ \frac{GM m}{r^2} &= m\, a \implies a = \frac{GM}{r^2}. \end{align}$$

So all objects will experience the same acceleration, provided they are the same distance from the bigger object. The mass of the falling object does not affect its motion. This is called the equivalence principle.

At the surface of the Earth, everything is $r\approx 6380\, \mathrm{km}$ from its center, and the acceleration of any object in free fall is about $9.8$ m/s$^2$. But if you were to go up significantly higher this would change. At $300$ km above the surface, which is about the orbital location of the International Space Station, the acceleration is about $8.9$ m/s$^2$.

At any height, everything falls at the same rate, but that rate may be slightly different from the constant you learn in school. But if you stay within a few kilometers of the surface the constant rate is good to better than 1%, which is fine for most applications.

Answer 3

Every single atom pulls every other atom with the same force (not exactly but good enough). If you double the weight of one body and triple the weight of the other, then you have twice as many atoms on one side pulling three times as many on the other side, so six times as much force. Twice the acceleration on the second body and three times as much as the first.

Now this is strange because a single atom has unlimited force if you put enough mass nearby. That’s not how a locomotive works, for example. It doesn’t have more power when you add more carriages. On the other hand, if an atom had a fixed amount of gravitational force then it could attract a single nearby atom with incredible acceleration.

Answer 4

Assume this situation:

A mass $~m_1~,m_2~$ both are at the start position $~R+h_0~$ and both are falling to the surface of the earth, there isn't any air drag.

The equation of motion for $~m_1~$

$$m_1\,\ddot h_1(t)=G\,\frac{m_1\,M}{(R+h_1(t))^2}$$

and for $~m_2~$

$$m_2\,\ddot h_2(t)=G\,\frac{m_2\,M}{(R+h_2(t))^2}$$

from here, the acceleration of mass $~m_1~$ is equal to the acceleration of mass $~m_2~$, hence both masses reach the earth surface an the same time (Galileo's free fall theory).

With constant force $~F~$

$$m_1\,\ddot h_1(t)=F\\ m_2\,\ddot h_2(t)=F$$

you obtain that $~\ddot h_1(t)\ne\ddot h_2(t)~$, but this contradicts the Galileo free fall theory.

Answer 5

Why does gravity cause objects to pull other objects with constant acceleration and not constant force ?

Science generally doesn't give very satisfactory answer to "why is the universe the way it is" questions. However, we can attempt a couple of answers here.

First we can use the anthropic principle - if gravity behaved in a significantly different way to how we observe it to behave, then stable structures such as galaxies and solar systems would not be possible, and intelligent life would not have evolved to ask the question. This doesn't prove that gravity has to behave as it does, but it does suggest that gravity has to behave as it does in a universe that supports intelligent life.

Secondly, we can use Einstein's theory of general relativity, which says that gravity behaves the way it does because it is not in fact a force, but is actually the warping of spacetime in the neighbourhood of a concentration of mass/energy. And this warping of spacetime must have the same effect on the dynamics of all objects at a given location, independent of their mass (as long as their mass is small compared to whatever is warping spacetime). Of course, this then leads to the question "but why is general relativity the way that it is ?".

Answer 6

I think gravity may not be a force per se. If it were a force then it could be expressed as a quanta. Unfortunately, Einstein's theories cannot be reconciled with quantum theory and quantum theory is as yet not able to deal with gravity.

Einstein theories were all about the warping of space/time. Objects of differing masses all move through warped space time with the same acceleration.

Eistein's equations are notoriously difficult to follow and understand, but they do quantify and thereby confirm our everyday experience of accelerating bodies in a gravitational field or more correctly expressed as warped space time.