Why do things accelerate due to gravity at the same rate?

[Josh S Thompson]

If you use Newtons gravity equation; F=M1*M2/d^2. The force is dependent on the mass of two bodies so how could the mass cancel? Also, for example If I had a bowling ball and a ball the mass of the sun and droped them 100 feet above Earth The two balls would not accelerate at the same rate because one ball is so massive that it have much more force.

 

[shortyjat]

Things don't accelerate due to gravity at the same rate. What you're referring to, I think, is when people say that dropping a bowling ball and a marble from a building, both will hit the ground at the same time. People say this because the mass of both the bowling ball and the marble are negligible in comparison to the mass of the earth. Essentially, the bowling ball IS accelerating faster, however the difference is so small that it doesn't really matter.

 

[ShayanJ]

Using Newton's 2nd law and his law of acceleration, we have $m\vec a=-G \frac{m M}{r^2}\hat r \Rightarrow \vec a=-G \frac{M}{r^2}\hat r$. So the acceleration of an object due to gravity doesn't depend on the mass of the object, so its the same for all objects.
Actually the reason is that the gravitational and inertial masses of an object are equal. Otherwise we couldn't cancel them above and acceleration would depend on the mass of the object.
This is called weak equivalence principle.

 

[davenn]

Drakkith did an awesome post on this topic a day or 2 ago in this thread ...
https://www.physicsforums.com/threads/object-falls.815279/page-2

post #22

work your way through that and see if you still have any questionsDave

 

[Drakkith]

Things don't accelerate due to gravity at the same rate. What you're referring to, I think, is when people say that dropping a bowling ball and a marble from a building, both will hit the ground at the same time. People say this because the mass of both the bowling ball and the marble are negligible in comparison to the mass of the earth. Essentially, the bowling ball IS accelerating faster, however the difference is so small that it doesn't really matter.

This is not correct. In the absence of air resistance, both the bowling ball and the marble have the exact same acceleration. What's different is that the Earth accelerates at a higher rate under the influence of the bowling ball's gravity than the marble's.

 

[shortyjat]

This is not correct. In the absence of air resistance, both the bowling ball and the marble have the exact same acceleration. What's different is that the Earth accelerates at a higher rate under the influence of the bowling ball's gravity than the marble's.

Gotcha. Thanks!

This is not correct. In the absence of air resistance, both the bowling ball and the marble have the exact same acceleration. What's different is that the Earth accelerates at a higher rate under the influence of the bowling ball's gravity than the marble's.

Gotcha. I checked out your explanation in another thread and it was much more thorough than the explanation given to me.

 

[Josh S Thompson]

This is not correct. In the absence of air resistance, both the bowling ball and the marble have the exact same acceleration. What's different is that the Earth accelerates at a higher rate under the influence of the bowling ball's gravity than the marble's.

Thank you I agree with this, so essetially you can say if the objects are in the same location they accelerate at the same rate because Earth accelerates to one point. But if the objects are far away there would be a difference in the perceived acceleration i.e.one object hits the ground first.

 

[davenn]

But if the objects are far away there would be a difference in the perceived acceleration

what do you mean by this ??

 

[Drakkith]

Thank you I agree with this, so essetially you can say if the objects are in the same location they accelerate at the same rate because Earth accelerates to one point. But if the objects are far away there would be a difference in the perceived acceleration i.e.one object hits the ground first.

I think you're saying that a larger object dropped from a large distance would hit the ground first because the Earth accelerates faster towards it. If we were to set up two different situations where M₁ and M₂ are dropped from identical distances, then yes, that would be correct.

 

[ZapperZ]

If you use Newtons gravity equation; F=M1*M2/d^2. The force is dependent on the mass of two bodies so how could the mass cancel? Also, for example If I had a bowling ball and a ball the mass of the sun and droped them 100 feet above Earth The two balls would not accelerate at the same rate because one ball is so massive that it have much more force.

Read the FAQ:
https://www.physicsforums.com/threads/why-is-acceleration-due-to-gravity-a-constant.511172/

Zz.

 

[Josh S Thompson]

what do you mean by this ??

If you in the middle of the objects that are far away from each other but the objects are equal distance above Earth it would look like the heavier object accelerates faster and it would hit ground first

 

[Drakkith]

If you in the middle of the objects that are far away from each other but the objects are equal distance above Earth it would look like the heavier object accelerates faster and it would hit ground first

Only if you dropped them separately and measured the impact time for each one. If you drop them together then they still hit the ground at the same time.

 

[Josh S Thompson]

Only if you dropped them separately and measured the impact time for each one. If you drop them together then they still hit the ground at the same time.

what you mean, all other things constant the heavier object would hit the ground first if you dropped them together because there is more force between them.

 

[davenn]

what you mean, all other things constant the heavier object would hit the ground first if you dropped them together because there is more force between them.

did you not read Drakkith's post in that thread I linked to in post #4 ?

if you did you shouldn't still be asking these same questions over and over

Again ...
negating atmospheric drag ( resistance) 2 objects, regardless of difference in mass, if dropped at the same time, will fall and hit the ground at the same time
They are BOTH being subject to the same gravitational force
Their individual masses are irrelevant

Dave

 

[Josh S Thompson]

did you not read Drakkith's post in that thread I linked to in post #4 ?

if you did you shouldn't still be asking these same questions over and over

Again ...
negating atmospheric drag ( resistance) 2 objects, regardless of difference in mass, if dropped at the same time, will fall and hit the ground at the same time
They are BOTH being subject to the same gravitational force
Their individual masses are irrelevant

Dave

I'm not trying to read a book bro

 

[Josh S Thompson]

did you not read Drakkith's post in that thread I linked to in post #4 ?

if you did you shouldn't still be asking these same questions over and over

Again ...
negating atmospheric drag ( resistance) 2 objects, regardless of difference in mass, if dropped at the same time, will fall and hit the ground at the same time
They are BOTH being subject to the same gravitational force
Their individual masses are irrelevant

Dave

Ok this is what I am saying

"The general form of the equation for the force of gravitation is: F = GM1M2/r2 This means that the force, which is the same magnitude for both objects" (Drakkith)

if the force is acting on both objects equally why can't you say the F acting on the Earth from gravity is MA, then set it equal to the force of gravity between two objects

MA=M*m*G/(r^2) = A = m*G/(r^2)
M=mass of earth
A=acceleration of earth
m=mass of an object
r=distance
G=constant

Then the acceleration of Earth would be dependent on the mass of an object
and you could say a heavier object does fall to Earth faster
because we are on Earth and it would be hard to notice Earth moving.
And the Earth would touch the heavier object first

 

[davenn]

Then the acceleration of Earth would be dependent on the mass of an object
and you could say a heavier object does fall to Earth faster
because we are on Earth and it would be hard to notice Earth moving.
And the Earth would touch the heavier object first

but as your have been told repeatedly ... the mass of the object isn't considered and doesn't need to be

PLEASE READ that post from Drakkith

 

[Josh S Thompson]

but as your have been told repeatedly ... the mass of the object isn't considered and doesn't need to be

PLEASE READ that post from Drakkith

I did read it and the part where he plugs in the F with MA, I say why can't you say the force acting on the Earth is equal to MA

M=mass of earth
A=acceleration of earth
m=mass of an object
r=distance
G=constant

M*m*G/(r^2) = force acting on earth

MA = definition of a force

MA=M*m*G/(r^2) =

A = m*G/(r^2)

Read that Dave

 

[Drakkith]

Ok this is what I am saying

"The general form of the equation for the force of gravitation is: F = GM1M2/r2 This means that the force, which is the same magnitude for both objects" (Drakkith)

if the force is acting on both objects equally why can't you say the F acting on the Earth from gravity is MA, then set it equal to the force of gravity between two objects

MA=M*m*G/(r^2) = A = m*G/(r^2)
M=mass of earth
A=acceleration of earth
m=mass of an object
r=distance
G=constant

Then the acceleration of Earth would be dependent on the mass of an object
and you could say a heavier object does fall to Earth faster
because we are on Earth and it would be hard to notice Earth moving.
And the Earth would touch the heavier object first

Sure. But here's where we have to be careful with terminology and word choice. When you say that the heavier objects falls to Earth faster, most of us interpret that to mean that the acceleration of the object under the force of Earth's gravity is more for M₁ (heavier mass) than for M₂ (lighter mass), which is NOT true. Both objects experience the same acceleration towards the Earth from Earth's gravity.

Now, if you want to account for the slight acceleration of the Earth towards the objects, then for clarity's sake we need to use an inertial coordinate system to compare everything to and not the accelerating frame of the Earth.

I did read it and the part where he plugs in the F with MA, I say why can't you say the force acting on the Earth is equal to MA

It is equal to MA. For a 200 kg mass placed just above the surface of the Earth, the force exerted on the Earth by the mass is 1962 Newtons and the acceleration would be 3.29 x 10-22 m/s². So 1962 N = (5.97 x 10²⁴ kg) * (3.29 x 10-22 m/s²). F=MA.

 

[paulfr]

The bowling ball feels a greater Force from the Gravitational Mass of the Earth
than the marble as calculated with Newton's Universal Law of Gravitation.
But it also has more Inertial Mass, the tendency to resist a change in motion.

The two effects balance so that the bowling ball accelerates at exactly the same
rate as the marble.

I believe the argument that they accelerate slightly differently due to
their much smaller mass than that of the Earth is invalid.
Newton's Universal Law + Newton's 2nd Law F=ma show their
accelerations to be identical and independent of their mass.

 

[ZapperZ]

The bowling ball feels a greater Force from the Gravitational Mass of the Earth
as calculated with Newton's Law of Gravitation.
But it also has more Inertial Mass, the tendency to resist a change in motion.

The two effects balance so that the bowling ball accelerates at exactly the same
rate as the marble.

While this is a perfectly acceptable qualitative explanation to give a conceptual understanding, note that to be able to say "... the two effects balance..." requires a quantitative explanation. There is no way to know that they EXACTLY "balance" each other such that the acceleration remains the same until one proves it via mathematics.

Zz.

 

[Nugatory]

I believe the argument that they accelerate slightly differently due to
their much smaller mass than that of the Earth is invalid.
Newton's Universal Law + Newton's 2nd Law F=ma show their
accelerations to be identical and independent of their mass.

It depends on whether you're considering the acceleration relative to the surface of the earth, or relative to a fixed point in space. These are the same thing if the Earth doesn't move, which will be the case if the gravitational force is small enough that its effect on the Earth is negligible.

 

[Buckleymanor]

This is not correct. In the absence of air resistance, both the bowling ball and the marble have the exact same acceleration. What's different is that the Earth accelerates at a higher rate under the influence of the bowling ball's gravity than the marble's.

Being a bit of a pedant I don't reckon this is completely true. With the absence of air resistance the marble is ever so slightly attracted to the bowling ball and the bowling ball is accelerating at an even slower rate towards the marble.
The three body problem springs to mind, inconvenient but true.http://www.askamathematician.com/2011/10/q-what-is-the-three-body-problem/

 

[Nathzie Enungan]

wow... thank you to all of your ideas everyone. I understand it now. :)

 

[Drakkith]

Being a bit of a pedant I don't reckon this is completely true. With the absence of air resistance the marble is ever so slightly attracted to the bowling ball and the bowling ball is accelerating at an even slower rate towards the marble.

Of course. I was ignoring the attraction the two balls had to each other because it doesn't change how fast the two accelerate towards the Earth or vice versa.

 

[Buckleymanor]

Of course. I was ignoring the attraction the two balls had to each other because it doesn't change how fast the two accelerate towards the Earth or vice versa.

It does the marble takes a longer slightly more arced (geodesic) path towards the Earth than the bowling ball.

 

[Drakkith]

It does the marble takes a longer slightly more arced (geodesic) path towards the Earth than the bowling ball.

If we're going to get this detailed then we might as well bring back air resistance. The point was that the mass of the object does not affect the acceleration of the object under gravity.

 

[Buckleymanor]

If we're going to get this detailed then we might as well bring back air resistance. The point was that the mass of the object does not affect the acceleration of the object under gravity.

Of course it does, let's go back to the O.P which mentions a ball with the mass of the Sun and a bowling ball.
They don't have the same acceleration nor does the marble and the bowling ball which was introduced in a later post.
Air resistance was not introduced by the O.P.
Some of these effects are negligible but what is the point of totally ignoring them.
I did mention I was a bit of a pedant but it is o.k to go into some detail if you want to get a clearer picture of what happens.

 

[Drakkith]

Of course it does, let's go back to the O.P which mentions a ball with the mass of the Sun and a bowling ball.
They don't have the same acceleration nor does the marble and the bowling ball which was introduced in a later post.

Yes they do. That's exactly what we've been showing here in this thread.
If you're thinking of geodesics as in General Relativity geodesics, then that is simply beyond the scope of this thread, as we are discussing classical gravity. General relativity would be far beyond the OP's knowledge level.

 

[epenguin]

Using Newton's 2nd law and his law of acceleration, we have $m\vec a=-G \frac{m M}{r^2}\hat r \Rightarrow \vec a=-G \frac{M}{r^2}\hat r$. So the acceleration of an object due to gravity doesn't depend on the mass of the object, so its the same for all objects.
Actually the reason is that the gravitational and inertial masses of an object are equal. Otherwise we couldn't cancel them above and acceleration would depend on the mass of the object.
This is called weak equivalence principle.

Er, only that's not a reason is it? It's not a scientific explanation of any fact. It's nothing but a restatement of the fact (that "the acceleration of an object due to gravity doesn't depend on the mass of the object, so its the same for all objects.").

All it does is fix things right with the taught concepts and principles of mechanics? - (that I'm never sure don't contain some circularity or unobservables)

 

[ShayanJ]

Er, only that's not a reason is it? It's not a scientific explanation of any fact. It's nothing but a restatement of the fact (that "the acceleration of an object due to gravity doesn't depend on the mass of the object, so its the same for all objects.").

All it does is fix things right with the taught concepts and principles of mechanics? - (that I'm never sure don't contain some circularity or unobservables)

If you read OP's question, you'll see that it wasn't as obvious for him\her.

 

[pbuk]

What we have here is a failure to communicate - Josh S Thompson you do not express yourself clearly, and you don't listen to what people say. But to be fair, some of the people providing answers aren't listening to what YOU say either:

If I had a bowling ball and a ball the mass of the sun ...

In this case we clearly do need to take into account the acceleration of the Earth which. So to answer your original question (actually it wasn't a question, but let's ignore that for a moment,

The two balls would not accelerate at the same rate

From the point of view of an independent observer (more technically, an inertial frame of reference), the two balls would each accelerate towards the Earth at $\frac{Gm_E}{r^2}$ where $m_E$ is the mass of the Earth. However, just as the acceleration of the ball depends only on the mass of the Earth (and the distance between them), the acceleration of the Earth depends only on the mass of the ball. In the case of the bowling ball this is negligible, but for a ball the mass of the Sun it is much larger than that of the Earth - this is an important property of a celestial body known as its standard gravitational parameter: you can find some of these (including for the Earth and for the Sun) in this Wikipedia article.

So whereas if we drop a bowling ball from 100 ft we can ignore the motion of the Earth and calculate that the ball accelerates at about 9.8 ms^-1 and hits the surface in 2.5 s, if it were possible to place an object the size of a bowling ball but the mass of the Sun 100 ft above the surface of the Earth and release it, we can ignore the motion of the solar mass object and calculate that the Earth would accelerate at about 3,300,000 ms^-1 and collide in 0.0043 s.

So from the point of view of an observer on the surface of the Earth, the Sun would indeed accelerate 330,000 times faster than the bowling ball.

 

[pbuk]

It does the marble takes a longer slightly more arced (geodesic) path towards the Earth than the bowling ball.

That doesn't affect the downwards component of acceleration, or the time it takes to collide.

 

[TurtleMeister]

One of the reasons (and possibly the main reason) people have a hard time understanding the universality of free fall is that they confuse relative acceleration (acceleration relative to one of the accelerating bodies) with acceleration relative to the inertial frame of reference. This is what Nugatory was referring to in his post #22. The UFF is only valid when using an inertial frame of reference.