How Strong is Gravity?

1. Jun 19, 2015

robbertypob

Something is confusing me...

1. If I drop a cannonball and a feather in a vacuum they will fall to Earth at the same rate. The mass and size of the objects do not appear to be affecting the 'strength' of the gravitational force acting upon it.

2. The Earth's gravity is stronger than the Moon's, because the Earth is more massive.

How can both these statements be true? I must be missing something. It looks like mass doesn't matter in the first statement but in the second statement it does?!

2. Jun 19, 2015

phinds

What you are missing is that the feather moves the Earth less than the cannonball. We don't normally think of the Earth as moving in that transaction, but it does (albeit in a a trivial and in all likelihood unmeasurable amount). THAT's where the mass matters. It would be more obvious with the moon. If the moon were not orbiting the Earth, but were to magically appear (defying all laws of physics) in its current position but not orbiting, it and the Earth would fall towards each other, but the Earth would move more slowly than the moon and they would meet up closer to the starting point of the Earth than the starting point of the moon (LOTS closer, actually).

3. Jun 19, 2015

Staff: Mentor

Mass does matter, but it takes more force to accelerate the cannonball than the feather. If they accelerate at the same rate, the forces must be unequal. The relationship if F=ma, force equals mass times acceleration. Therefore F/m for the cannonball is the same as F/m for the feather.

Does that make sense to,you?

4. Jun 19, 2015

robbertypob

OK, let me see...

So, gravity is calculated as Force = Mass x Acceleration. The cannonball has a greater mass but that means it has a lower acceleration. The feather has a lesser mass but that means it has a higher acceleration.
Any differences in mass are offset by the difference in acceleration, meaning they appear to fall at exactly the same rate.

Is that right? Have I understood it?

5. Jun 19, 2015

Staff: Mentor

Nope. F=ma is Newton's second law. It works for any force, not just gravity

"difference in acceleration, meaning they appear to fall at exactly the same rate.". That sentence is self contradictory. If there are differences in acceleration, then they don't fall at the same rate. The feather and the cannonball have the same acceleration.

6. Jun 19, 2015

robbertypob

Thanks for explaining but I just don't understand.

I don't understand how greater mass means greater gravity and yet the cannonball falls at the same rate as the feather.
If the cannonball is more massive than the feather then it must have more gravity and therefore the gravitational attraction between the Earth and the cannonball is stronger than between the Earth and the feather. It just seems to me that the cannonball should be falling more quickly than the feather, even if that difference is marginal.

7. Jun 19, 2015

phinds

Think of inertia. It's harder to get the cannonball moving, which offsets its "greater gravity" as you put it, so it falls at the same rate. It's all there in the math but perhaps the inertia consideration will make it more understandable to you.

8. Jun 19, 2015

A.T.

Two cannonballs have greater mass than one cannonball. Should two cannonballs fall faster if glued to one object then when they are separate?

9. Jun 19, 2015

robbertypob

Thanks for helping me to understand this.

I think I get it now - the cannonball has a greater mass but a slower acceleration and therefore falls at the same rate.

10. Jun 19, 2015

phinds

No, rate and acceleration are the same thing. It does NOT have a slower acceleration (or a faster one) or else it would not fall at the same rate as the feather.

11. Jun 19, 2015

robbertypob

My brain hurts! I need to read up on some more basic concepts before trying to comfortably understand this.

12. Jun 19, 2015

Bandersnatch

Let's get back to the basics here.

Acceleration (the $a$ in $F=ma$) IS the rate of falling. The $m$ in $F=ma$ tells you that the higher the mass of the object whose motion you're considering, the higher force is required to move it. You'd normally call this $m$ inertia.

So for two objects of different masses, say $m_1$ and $m_2$, to fall at the same rate $a$, the force $F$ must be higher or lower proportional to the difference in masses.
Say, $m_1=1kg$ and $m_2=2kg$. You need twice the force for $m_2$ to accelerate it at the same rate as you would $m_1$.

This works for all forces. The $F$ in $F=ma$ doesn't tell you anything about which kind of force is it.

Now, look at the force in question. The force of gravity (law of gravity) is $F_G=GMm/R^2$. This force is a force that always acts between two bodies. These two bodies are represented by their masses $M$ and $m$. $G$ is a constant that is not very important here, $R$ is the distance between the bodies.

Notice, that the force of gravity increases if you increase either of the masses.

So, while $F=ma$ means that mass increases inertia, making it harder to accelerate, the law of gravity tells you that the force on an object being accelerated increases by the same proportion as inertia.

You can combine the two equations to get:
$ma=GMm/R^2$
You divide both sides by $m$ to get the equation of acceleration:
$a=GM/R^2$
which is independent of the mass $m$! It still is dependent on the mass $M$. So all objects, regardless of what exactly their mass $m$ is, will fall at the rate $a$ determined solely by the mass of the body $M$.

In all of this, an important assumption is made, that $M$ is much, much greater than any mass $m$ whose motion you're looking at.
This is to ensure that you don't have to worry about the mass $M$ being accelerated towards the smaller mass $m$. This is a good approximation for dropping spoons, feathers and cannon balls onto planets and moons, but in principle the Earth does get accelerated to a cannon ball faster than to a feather, and the two don't fall at exactly the same time as a result.
However, this kind of analysis is more involved, so until you understand the basic one described above, leave it be. All you need to remember is that there is a deeper level to falling at the same rate, that is negligible for small object falling onto a huge one.

13. Jun 19, 2015

robbertypob

Thank you for such a detailed and understandable explanation! So the actual force of gravity is greater on the cannonball than the feather - that is fascinating to me.

14. Jun 19, 2015

rumborak

I get the impression that a good amount of confusion comes from using the very vague term "rate". In reading replies I've seen people mean both the acceleration and the speed by it.
We should stick to the much less ambiguous terms "speed" and "acceleration".

15. Jun 19, 2015

rumborak

BTW, this might make it a bit more intuitive why it is that way:
Imagine two feathers in a vacuum, a meter apart from each other. You drop them simultaneously. Obviously, they will come to the ground at the same speed. Now, bring the two feathers to each other, i.e. glue them together. Drop them now. What would happen? The same thing really. It's not as if one feather is trying to fall faster than the other, dragging the other with it. So, the two-feather blob takes the same time to drop as the two separate feathers. And of course twice the force was applied to the two-feather object, after all it's two of them now.
Well, now take 1,000 feathers and bunch them all together, and you got yourself a cannonball.

16. Jun 19, 2015

robbertypob

Yes I see what you mean!

Gravity is really interesting to me. It just seems to behave in a bizarre way (i.e. the force is actually stronger on the cannonball than the feather), although of course it makes sense when you look at the equations. Do we understand how gravity works at a molecular level? What specifically is happening to attract the two masses towards each other?

17. Jun 19, 2015

rumborak

Let me ask you, why is it bizarre? After all, both the feather and the cannonball are made of the same particles, just a different amount. Each particle on its own experiences the gravitational pull, and so each of them falls with the same acceleration. It doesn't matter whether those particles stick together or not.

18. Jun 19, 2015

Bandersnatch

Pfft. Such lack of accuracy. On PF, of all places!
A broiler chicken has some 9000 feathers on average, which altogether weigh about 74 grams.
The lightest cannon ball according to Borgard's ordnance standards in the 18th century British Navy weighs 4 pounds.
That means you need on the order of 200 000 feathers for a cannonball.

19. Jun 19, 2015

robbertypob

Ah yes it's helpful for me to think of it in that way - gravity is actually acting on each particle individually. It doesn't make any difference at all (at the level of cannonballs and feathers) what the overall mass of the object is. If the mass is greater then the force of gravity will be greater.

I'm sure it's not bizarre but it seems so to me because I can't think of any other force that behaves in that manner. If I were to have a marble in one hand and a baseball in the other and threw them both at a wall I would have to think very carefully about how much force to apply to both if I wanted them to reach the wall at the same time due to the differences in mass, but this doesn't matter with gravity. They will arrive at their destination at the same time. Wow!

20. Jun 19, 2015

phinds

Nitpicker ... them's a lot of chickens

21. Jun 19, 2015

A.T.

Indeed, the proportionality to mass is usually a property of inertial forces, which are coordinate effects not interactions. That’s why in General Relativity gravity is modeled as such a coordinate effect:

22. Jun 19, 2015

robbertypob

That is so interesting. I'm just about getting my head around the Newtonian idea of gravity... Einstein's bending spacetime may have to wait until I've had some coffee.

23. Jun 19, 2015

olgerm

Fgravitational=m1*m2*G/r2
a=F/m

Cannonball falling down onto Earth:
Fcannonball=mcannonball*mEarth*G/r2
acannonball=Fcannonball/mcannonball=Fcannonball=mcannonball*mEarth*G/(r2*mcannonball)=Fcannonball=mEarth*G/r2
(acannonball does not depend on the mass of the cannonball)

Leather falling down onto Earth:
FLeather=mLeather*mEarth*G/r2
aLeather=FLeather/mLeather=FLeather=mLeather*mEarth*G/(r2*mLeather)=FLeather=mEarth*G/r2
(aleather does not depend on the mass of the Leather)

Fcannonball≠FLeather
aleather=acannonball=mEarth*G/r2

a is acceleration. Acceleration is rate of change of velocity.
r is distance between interacting bodies centers of mass(from falling body to center of Earth). r≈rEarth radius≈6371000*m
G ≈6.67*10-11 https://en.wikipedia.org/wiki/Gravitational_constant

Last edited: Jun 19, 2015
24. Jun 19, 2015

Bandersnatch

I'd say it is, and on a deep level. There are forces that can be described with an almost identical equation as the force of gravity. Like the Coulomb force between two charged particles: $F_C=k Qq/R^2$, where $k$ is again some constant, like $G$, and the two q's are electrical charges.

This is a very general kind of equation - the $R^2$ tells you how the force propagates in three dimensions, the constant tells you how to correctly calculate units between the two sides of the equation (and how 'strong' the force is), and there are two 'charges', which are some properties of things whose interaction causes the force.

But when you link it to $F=ma$, you get: $ma=k Qq/R^2$, and the inertial mass from $ma$ certainly doesn't go away.

How is it, that the same quantity that determines inertia is acting as the 'charge' for the force of gravity? Why would the two be the same? We introduce them in school with the same symbol $m$, so their equality might seem natural, but we could very well call one by a different symbol. On a deeper level, up until relativity comes along, there's just no good reason for why they should be equal, and people had been somewhat mystified by this for a good while. There were efforts to measure whether the gravitational and inertial masses are really equal, and they concluded that indeed it seems so to a very high degree of accuracy.

As mentioned by A.T., the solution to this conundrum came with General Relativity, which did away with gravity being a force.
It can still be modelled as such for a wide range of regimes, though.

25. Jun 19, 2015

jerromyjon

That part isn't quite "figured out" yet. Gravity between 2 atoms is virtually undetectable and gets into very complicated models of quantum physics.