Unraveling the Mystery of Mass and Acceleration

[xWaldorf]

So, this may be a really stupid question, and I strongly feel as though I'm missing something here.
How can it be that objects of different masses have the exact same acceleration when mass is in fact resistance to acceleration?
And then, if in (a vaccum) I throw upwards M and m ( a bigger and a smaller mass respectively) will m reach the ground first or will they still reach the ground simultaneously?

 

[Gaussian97]

Well, it's true that mass is a measure of how difficult is to accelerate a body, but this doesn't prevent you to accelerate equally two different bodies, only you need to try harder with the massive one. That's exactly what gravity does.

Let me explain, of course with an equal force, the acceleration of a massive body will be higher(edit: lower) than a lighter body, that's what Newton's second law tells us; $\vec{a}=\frac{\vec{F}}{m}$. But notice that here there are two things that can contribute, one is the force and the other is the mass. That's why I've explicitly said that under an equal force the acceleration is higher in the massive body.

But gravity does not act as an equal force to everybody, Newton's gravitational law tells us that the more massive a body is, the more force will experiment due to gravity. It turns out that the two effects (more mass implies more force, but also less acceleration) cancel out, so the total acceleration due to gravity is exactly the same for all the bodies.
$$\vec{F}_G=m\vec{g}\Longrightarrow \vec{a}=\frac{\vec{F}_G}{m}=\vec{g}$$
where $\vec{g}$ is a constant that doesn't depend on $m$.

Then the second question is easy, if they have the same acceleration (and you threw them with the exact same velocity), they will arrive at the ground simultaneously.

 

[xWaldorf]

of course with an equal force, the acceleration of a massive body will be higher than a lighter body, that's what Newton's second law tells us; →a=→Fma→=F→m\vec{a}=\frac{\vec{F}}{m}.

Having an equal F in both cases, wouldn't plugging a greater mass in the equation results in a higher acceleration for the lighter body (because the denominator is greater)?

 

[PeroK]

Summary:: If mass is a measure of a bodys resistance to acceleration, how come objects with different masses accelerate the same due to gravity?

So, this may be a really stupid question, and I strongly feel as though I'm missing something here.
How can it be that objects of different masses have the exact same acceleration when mass is in fact resistance to acceleration?
And then, if in (a vaccum) I throw upwards M and m ( a bigger and a smaller mass respectively) will m reach the ground first or will they still reach the ground simultaneously?

I think there are several parts to your question. (Although I've been beaten to it, I see!)

First, inertial mass is a measure of a body's resistance to acceleration. As in Newton's second law:
$$F = ma$$
This equation relates the force applied to a body to its acceleration.

Second, gravitational mass is a measure of the gravitational attraction of a body. As in Newton's law of gravitation: $$F = \frac{GMm}{r^2}$$
This equation gives the force of attraction between two bodies of masses $M$ and $m$, separated by a distance $r$. $G$ is the universal gravitational constant.

It has long been established experimentally that inertial and gravitational mass are the same. For example, if we take $M$ to be the mass of the Earth, then the gravitational acceleration of a mass $m$ (due to the Earth's gravity) is:
$$a = \frac F m = \frac{GM}{r^2}$$
Which is independent of the mass $m$. This can be tested by dropping two objects of different masses and checking that they fall at the same rate. Note that the masses have to be dense enough so that air resistance isn't an issue. In a vacumm chamber, however, you can see this applies to all objects.

The reason is that if you increase mass $m$, then both the gravitational force (between it and the Earth) and its resistance to acceleration increase by the same amount and cancel out.

 

[Gaussian97]

Having an equal F in both cases, wouldn't plugging a greater mass in the equation results in a higher acceleration for the lighter body (because the denominator is greater)?

Yeah, sure, I don't know what I was thinking

 

[xWaldorf]

hahahahaha thanks for the lucid answers though!