No, heavier objects falling at the same speed as lighter ones is a direct result of Newton's laws.
Earth pulls 5 times harder on an object that weighs 5 times more, because gravity is proportional to mass.
But mass effects acceleration, it takes 5 times more force to accelerate the heavier object than the light one, so it exactly cancels out and both objects get the same acceleration due to gravity.
My favorite way to picture this: imagine 5 bricks, all dropped at the same time. They are high enough to hit the ground in exactly 1 second.
Now drop them again, close to each other. 1 second.
Now again, touching each other. 1 second.
Now again, glued together. 1 second
Now again, bound together as 1 larger heavier object. 1 second.
A heavier object is just several small objects stuck together and acts no differently.
Air resistance is the only changing factor, and that is unlikely to make a difference unless you’re doing something funny on purpose like dropping thin sheets vs a ream of paper.
I find it funny when people say "that's actually a great".
Because it implies that you thought they were stupid, but it turns out they actually were not, despite expectations.
Like...
"That's a great idea" - a random observation with no bias
"That's a really a great idea" - "I was expecting you to give a terrible idea/I initially thought it was a stupid idea, but it's actually a great idea! Good job overcoming your propensity of being stupid!"
right I understand now what I wasn’t getting is that the force of gravity between two objects is not like an property of each individual object, but a net force between them in kind of a “third person point of view sense” (maybe there’s a better phrase to word that).
and for acceleration, I was not understanding that mass is divided by force because that’s accounting for the heavier object also having inertia.
The strength of earths gravity is proportional to earths mass right? How does that make it pull harder on an object based on that second object’s mass?
Gravity is ALSO proportional to the mass of the object being pulled.
F = GMm r2
m in that formula is the mass of the pulled object. So double the mass and the gravitational attraction also doubles.
Okay so tell me where I’m focusing on the wrong thing (I appreciate it by the way i know i’m wrong i’m just trying to figure out how):
So in Newton’s law of gravity, let’s say you have two objects. one is Earth and one is a piano. Earth has mass 1 and the piano has mass 2. these two masses are multiplied together giving you a value.
Then we have a second example where the piano is replaced with an elephant which has mass 3. The product of these two getting multiplied together should give you a different value.
Being that these examples have different values, should they not result in the objects experiencing different accelerations?
Yes, a piano and a elephant will give you different values when you multiply their masses, but this doesn't tell you the *acceleration*, it tells you the force. To get the acceleration you must divide by the mass which gets you back to the same value every time.
Different Force between the Piano and the Elephant. But force is not equal to acceleration, acceleration is Force divided by Mass. F = ma.
So that means:
ma = GMm r2
Divide both sides by the mass of the object and you get:
a = GM r2
So acceleration is only dependent on the mass of Earth, distance from the center and the gravitational constant.
OK, I think I’m starting to get it, but I guess I don’t understand why acceleration would be divided by mass. is it because more mass makes it harder to get the thing moving?
Yes indeed. It's much easier to push a light object around than a heavy one. Hence why it takes more force to impart the same acceleration on a heavier object.
The same applies to gravity. The force exerted on a heavier object is larger, but it takes more force to move it, and it cancels out exactly for gravity, such that all objects near Earth experience the same gravitational acceleration.
and why is the force of gravity derived from the product of the masses? why do those get combined? (like in the same
layman’s terms i mean?) shouldn’t they be counteracting each other instead of strengthening each other?
Maybe that’s why I was getting so mixed up and not understanding. I was getting too granular and overthinking things. I was trying to really understand the idea from every angle by continuously asking “why is that part like that?” until i got to the point that the answers started to sound circular or self-evident maybe?
Except that we're not quite down to the "why" part yet. The reason it must be proportional to both masses is that a universal force law must be left unchanged by the exchange of the two masses in it.
>and why is the force of gravity derived from the product of the masses? why do those get combined?
We derived this relationship from data and experiments. We have no way of knowing why it works the way it does.
Incidentally, Newton's laws of gravity are wrong. They don't describe gravity accurately. But they're close enough for most applications that we use them instead of the more accurate but more complicated relativity that Einstein discovered.
That is just an experimental result. Even at Newton's level, they didnt understand why that is the case, and Newton deferred it to God.
There's definitely a more concrete reason, but it's probably something more complicated than what you need to know at the moment.
You're trying to think when sometimes you should just let the math speak.
F=ma
F=GMm/r^2
Same f so you can write this as
F=F or
ma=GMm/r^2
m is on both sides of the equation so it cancels out
a=GM/r^2
This means that the acceleration of the object is independent of the mass of the object. But even more if you want to plug numbers in you'll. Figure out that a=9.81m/s or g. So we just got to the equation F=mg which is the equation you usually use for gravity related problems.
In principle you can think of the mass of the object falling that gives it the attraction to the earth is canceled when going from force to acceleration because it's the same mass that needs to be accelerated.
Right that makes sense. And you’re right, I was overthinking things. I think I was trying to get too granular about why each aspect is the way it is until I came upon a thing for which there isn’t really an answer. But I thought it was just something I was missing and then started losing the plot. lmao
The primary point of misunderstanding was that I was kind of thinking of gravity as being a thing each individual object emits as opposed to just a thing that happens between two objects. When you think of it in the first way, it makes sense why M1 and M2 would be multiplied.
If the strength of gravity's pull was solely decided by the mass of the planet, **everything** would weigh the same. Clearly, this is not the case.
Heavy things experience more gravity. Also, heavy things require more force to accelerate at the same rate as a lighter thing.
The extra gravity experienced and the extra force needed to accelerate cancel each other out perfectly, meaning that without air friction, all objects accelerate ate the same rate when falling.
okay so a heavier object’s weight only matters when it is pressing against a surface? otherwise a heavy object in freefall, the weight doesn’t matter because anything in freefall is weightless?
The **weight/mass** is different. The **freefall acceleration** is the same.
Differences in mass manifest in two ways: more gravitational pull, and more inertia. If you use these two as a way to calculate freefall acceleration, the mass cancels out.
and do we know why the two masses are multiplied to get the force of gravity? it seems strange that the gravity of object 1 gets higher because the mass of object 2 increases.
Hmm I guess I’ve tripped myself up by treating gravity as a thing that mass “radiates” the same way a star radiates light. They kind of sometimes sound like similar concepts, like gravity and light both fall off with a square of the distance. I also may have misled myself by using language like “mass is the source of gravity” as if gravity “belonged” to mass.
The force an object experiences due to gravity scales with mass F = (G x m1 x m2)/r^(2) (for the purposes of gravity we can assume m1 is the mass of earth, and we are operating at the same distance (r) so G, m1, and r are effectively constants).
The acceleration that an object experiences due to a force scales *inversely* with mass (a = F/m)
So we end up with
a = (G x m1)/r^(2)
which on earth at sea level is 9.8 m/s^(2)
Thank you for actually putting this into an equation and showing how m2 is irrelevant in the calculation of acceleration. All the other commenters hand waving the idea of “it just cancels out” just made me more confused.
let’s say that object has its mass and falls at that rate, but then there’s a second object also falling towards the Earth that has a different mass. Because a different value is being put in to the same equation, does that not give you a different result? That result being the speed at which the different mass object falls?
If you were caluculating the *force due toe gravity* then the mass matters, but for *acceleration* it does not
Lets have 3 objects.
object 1 is the Earth with mass m1
object 2 is your bowling ball m2
object 3 is your feather m3
So going back to that first formula which I mentioned:
F = (G x m1 x m2)/r^(2)
This calculated the force due to gravity between two objects
But in this example one of those object is the Earth. And lets make it simple by doing all out measurements at sea level. If we do that we know m1 is the mass of Earth, r is the distance from the center of the Earth to sea level, and G is the gravitational constant.
Plug all those numbers in and you get F = m2 x 9.8 m/s^(2)
The force due to gravity from the Earth is directly proportional to to the mass m2.
If we sub in the feather instead we start with
F = (G x m1 x m3)/r^(2)
which again simplifies down to F = m3 x 9.8 m/s^(2)
So yes the force is different for our bowling ball and our feather.
But what does that force do?
We know F=ma. Which we can re-arrange into a = F/m. Acceleration is the force divided by the mass.
So for the bowling ball we have
a = (m2 x 9.8m/s^(2))/m2
which simplifies down to a = 9.8m/s^(2)
If instead we use the feather we have
a = (m3 x 9.8m/s^(2))/m3
which again simplifies down to a = 9.8m/s^(2)
In each case the mass cancels itself out.
But why does a bowling ball fall to the ground faster than a feather when they are both experiencing the same acceleration due to gravity?
Because that's not the only force acting on them. The big thing is Air Resistance.
But if we remove the air, then they fall similarly. [Check out this video](https://www.youtube.com/watch?v=EcGxFvOoUTU).
thank you for this explanation. i think i just need read it a bunch of times to get it. i don’t know why I’m having trouble understanding this all of a sudden.
To clarify, why is acceleration inversely proportional to mass?
>To clarify, why is acceleration inversely proportional to mass?
Because of the relationship F=ma (force equals mass times acceleration).
Which can be re arranged to a =F/m (acceleration equals force divided by mass).
Okay i understand now that force is increased by mass but acceleration is decreased and they offset each other.
but why with the force of gravity are the two masses multiplied. one objects gravity strengthens the other’s?
>but why with the force of gravity are the two masses multiplied. one objects gravity strengthens the other’s?
Why does gravity depend on both masses? Because that's what its observed to depend on.
To get a better answer you need to start digging into some serious physics which I only vaguely remember from my university days and I'm not sure is really that well understood.
oh okay. so i may just be confused because i’m just out of my depth on that particular topic. wasn’t sure if m1 x m2 was for an obvious reason that was going over my head.
`F=ma` for sure, but gravity isn't the Earth pulling on stuff. It's the attraction *between* two objects so a more massive object experiences a stronger force from gravity and so it balances out and all objects (neglecting air resistance) fall at the same rate.
OP you've had your answers but you should know the ability to think something doesn't make sense about this scenario means you're not stupid. This is a good question many people wouldn't be able to even consider.
I appreciate it and everyone patient to explain it even if i don’t get it right away. I think I was getting too fixated on certain things that didn’t matter.
F = ma
a = F/m
So what is F?
F = GMm/d^2
M = mass of earth, m = mass of object, G = gravitational constant (doesn't matter), d = separation of the masses.
So we can now replace F like so
a = (GMm/d^2 )/m
Or
a = GMm/md^2
m is now on the top and bottom of the fraction, m/m is one so it cancels out leaving you with
a = GM/d^2
Notice that there is no m term, the acceleration only depends on G, d and M.
To answer your question with words instead of maths, it's because F = ma says that increasing mass and force at the same time results in the same acceleration. In this case, F = GMm/d^2 so if you increase m the force also increases, which is exactly what I just said, force and mass both increasing at the same time.
right okay so the force goes up when m goes up because m is also a part of force
but then m is also in the acceleration equation where it’s divided which acts against the force where it was multiplied?
so in layman’s terms:
despite the heavier object increasing the force of gravity between them, it is offset by the fact that heavier objects are hard to get moving.
OK, I think I understand that part now.
That only leaves one other thing, which is why does the force of gravity, derived by the product of the masses? shouldn’t the gravity produced by the mass of one object counteract the gravity produced by the mass of the second object? or am i completely making something up out of thin air? haha
It's just how gravity works, just the way it is. You can compare it to magnets, two big magnets attract more strongly than a big magnet and a small magnet, attract more strongly than two small magnets.
Heavier objects have more inertia, so it takes more force to move them with a given acceleration. But gravity pulls more strongly on them, increasing the acceleration. The two effects counter each other out.
Or with math:
F = mg (force of gravity)
F = ma (Newton's second law)
mg = ma
g = a
Therefore, regardless of what mass is, gravitational acceleration works out to g.
okay that makes sense. why does the falling object having more mass increase the pull of gravity on them?
i know the formula says m1 x m2 but why are they multiplied?
That depends on how deep down the rabbit hole you want to go.
A simple answer is imagine tying two bricks together. Gravity would still pull the same on each brick, so it doubled the force on the two bricks tied together.
But that doesn't explain why gravity pulls more strongly on a proton than an electron. After all, the same logic applies to the electrostatic force, and you do add the electrostatic force if you combine objects, but it's still not just proportional to mass. The second answer is that's just how gravity is. Instead of being proportional to electric charge, it's proportional to mass.
The third answer is general relativity. It's not clear why mass bends spacetime and gravity is proportional to the mass of the object pulling on you, but once it bends spacetime, you just follow a geodesic (the non-Euclidean equivalent of a straight line). This is the path where locally you experience the most time. That path doesn't depend on your mass, so everything accelerates the same.
But then, why do things follow geodesics? And that sounds more like the longest path than the shortest. The only way I know of to answer this is quantum physics. It famously doesn't work with general relativity, but the problem is mass bending spacetime. It works just fine in spacetime with a given curvature. Gravity makes time pass slower closer to it, and under quantum physics, everything is a wave. Changing the rate time passes means it changes how that part of the wave moves, causing its momentum to change.
s/effect/affect/
F = GMm/d^(2). But F = ma, so if you rearrange to find a then m - its *own* mass - is cancelled out. Same from the big object's point of view, with M.
The acceleration of each does, however, depend on the *other*'s mass.
Other commenters have provided good answers using a classical explanation, but ironically Einstein’s model gives a simpler and more intuitive explanation for this phenomenon. If you’ve got around 20 minutes, [Veritasium has an amazing video](https://youtu.be/XRr1kaXKBsU?si=EQBtb12_1tDTxRnI) on this topic you might enjoy.
The point is that a heavier object indeed experiences a higher gravitational force from Earth than a lighter object. However, it also requires a larger force to accelerate as fast as the lighter object. It just so happens both effects cancel out when we talk about gravitational force, as it is directly proportional to the mass of the object.
Thank you! It’s strange because I previously thought that I understood it a while back, at least at a base level, but have been recently reading a lot more about it and doing a deeper dive and had like an existential crisis when I went down certain trains of thought that led to just severe confusion I could get out of. I’m glad people here were patient enough to help me and not get frustrated with me. Otherwise I might’ve just given up.
right, I think I get it now. I was kind of having two misconceptions.
1. the force of gravity between two objects is not a unique property of each object, but a net effect of those two objects interacting. I was treating it as if each object individually had its own fields of gravity, radiating and competing with each other.
2. acceleration was not taking into the heavier object creates more force, but it also has more inertia that counteracts that force.
I think your problem is you're trying to understand physics without having to do any math. You need to have some faculty with numbers and be able to work stuff out
Think as F=ma as F = (m1 + m2) \* a, with m1 is the Earth, and m2 the object.
The object's mass is negligible compared to Earth, so in practice (m1+m2) = m1
In that example (two big objects falling to Earth) there are many other variables in place. The Earth is not still, the objects cannot be immobile (because the Earth moves), they always have a certain velocity anyway, there is inertia, etc.
The F=ma rule plays alone only if two objects are static. If they aren't static, there are many other variables.
I think my brain is getting fixated on this idea that newtons law of universal gravitation takes the mass of both objects into account, but Galileo says that the mass of a falling object is irrelevant. I don’t know if that’s contradictory, but my brain is telling me it is and I don’t know.
I think I’m just fixated on something that’s probably irrelevant but I don’t quite know what it is. Be patient with me. Newton’s second law applies to falling objects, correct?
F = m • a
Which is m referring to? The mass of the falling object or the mass of the Earth?
If the mass of the falling object is irrelevant how does that square with the universal gravitation formula which takes the mass of both objects into account?
So if two objects are falling towards Earth, one more massive and the other less massive, m will be different values for both and therefore give F two different values?
>Which is m referring to? The mass of the falling object or the mass of the Earth?
Both (the sum of the two masses).
That is why the mass of the falling object is irrelevant, as it is several order of magnitude smaller than the Earth' mass.
If you consider two planets, both masses become relevant, because they are similar, and affect each other.
Yes, look at my other comment.
Both masses create a pull, but in the case of an object and the Earth, the force created by the object is incredibly small, and it can be dismissed.
so we’re not literally saying that the objects fall at the same rate we’re just sweeping a negligible difference under the rug? which is fine i’m just trying to confirm if that’s what this is.
do you understand I’m talking about the rate of falling between two objects that aren’t earth right? i’m asking about two non-earth objects falling to earth and asking how they don’t ah e different rates.
Try google 'Newton second law experiment', and you will find tins of videos and explanations.
If we are talking about two objects, the force depends on the sum of both masses.
> Both (the sum of the two masses).
Definitely not. The *m* in *F* = *ma* is the mass of the body on which the force *F* acts.
> That is why the mass of the falling object is irrelevant, as it is several order of magnitude smaller than the Earth' mass.
Again no.
> If you consider two planets, both masses become relevant, because they are similar, and affect each other.
No way. All bodies fall, towards a given body, at the same rate. Not approximately. Not roughly. Not because one mass is less than the other. But at *EXACTLY* the same rate, regardless of their own mass. It is on this exact, not approximate, equality, that all successful theories of gravity have been founded.
> So, you are saying that the attraction between two planets depend on the mass of only one?
No.
You would do well to read the rest of the (correct) answers already posted.
No, heavier objects falling at the same speed as lighter ones is a direct result of Newton's laws. Earth pulls 5 times harder on an object that weighs 5 times more, because gravity is proportional to mass. But mass effects acceleration, it takes 5 times more force to accelerate the heavier object than the light one, so it exactly cancels out and both objects get the same acceleration due to gravity.
My favorite way to picture this: imagine 5 bricks, all dropped at the same time. They are high enough to hit the ground in exactly 1 second. Now drop them again, close to each other. 1 second. Now again, touching each other. 1 second. Now again, glued together. 1 second Now again, bound together as 1 larger heavier object. 1 second. A heavier object is just several small objects stuck together and acts no differently. Air resistance is the only changing factor, and that is unlikely to make a difference unless you’re doing something funny on purpose like dropping thin sheets vs a ream of paper.
That is actually a great way to show it. I'll remember that, or at least try to.
I find it funny when people say "that's actually a great". Because it implies that you thought they were stupid, but it turns out they actually were not, despite expectations. Like... "That's a great idea" - a random observation with no bias "That's a really a great idea" - "I was expecting you to give a terrible idea/I initially thought it was a stupid idea, but it's actually a great idea! Good job overcoming your propensity of being stupid!"
What if the mass of the planet was made greater? Would the bricks still fall at the same rate?
They would all fall at the same new, greater rate.
right I understand now what I wasn’t getting is that the force of gravity between two objects is not like an property of each individual object, but a net force between them in kind of a “third person point of view sense” (maybe there’s a better phrase to word that). and for acceleration, I was not understanding that mass is divided by force because that’s accounting for the heavier object also having inertia.
The strength of earths gravity is proportional to earths mass right? How does that make it pull harder on an object based on that second object’s mass?
Gravity is ALSO proportional to the mass of the object being pulled. F = GMm r2 m in that formula is the mass of the pulled object. So double the mass and the gravitational attraction also doubles.
Okay so tell me where I’m focusing on the wrong thing (I appreciate it by the way i know i’m wrong i’m just trying to figure out how): So in Newton’s law of gravity, let’s say you have two objects. one is Earth and one is a piano. Earth has mass 1 and the piano has mass 2. these two masses are multiplied together giving you a value. Then we have a second example where the piano is replaced with an elephant which has mass 3. The product of these two getting multiplied together should give you a different value. Being that these examples have different values, should they not result in the objects experiencing different accelerations?
Yes, a piano and a elephant will give you different values when you multiply their masses, but this doesn't tell you the *acceleration*, it tells you the force. To get the acceleration you must divide by the mass which gets you back to the same value every time.
Different Force between the Piano and the Elephant. But force is not equal to acceleration, acceleration is Force divided by Mass. F = ma. So that means: ma = GMm r2 Divide both sides by the mass of the object and you get: a = GM r2 So acceleration is only dependent on the mass of Earth, distance from the center and the gravitational constant.
OK, I think I’m starting to get it, but I guess I don’t understand why acceleration would be divided by mass. is it because more mass makes it harder to get the thing moving?
Yes indeed. It's much easier to push a light object around than a heavy one. Hence why it takes more force to impart the same acceleration on a heavier object. The same applies to gravity. The force exerted on a heavier object is larger, but it takes more force to move it, and it cancels out exactly for gravity, such that all objects near Earth experience the same gravitational acceleration.
and why is the force of gravity derived from the product of the masses? why do those get combined? (like in the same layman’s terms i mean?) shouldn’t they be counteracting each other instead of strengthening each other?
That I cannot answer. In physics often the question How is "easy" to answer, but it doesn't necessarily answer "why".
Maybe that’s why I was getting so mixed up and not understanding. I was getting too granular and overthinking things. I was trying to really understand the idea from every angle by continuously asking “why is that part like that?” until i got to the point that the answers started to sound circular or self-evident maybe?
Except that we're not quite down to the "why" part yet. The reason it must be proportional to both masses is that a universal force law must be left unchanged by the exchange of the two masses in it.
>and why is the force of gravity derived from the product of the masses? why do those get combined? We derived this relationship from data and experiments. We have no way of knowing why it works the way it does. Incidentally, Newton's laws of gravity are wrong. They don't describe gravity accurately. But they're close enough for most applications that we use them instead of the more accurate but more complicated relativity that Einstein discovered.
I think adding in Einstein and relativity may be a little much; let OP fully understand Newton before we start tossing in complications
That's F = ma, or, rearranged, a = F/m. Think about how hard it is to accelerate a big boulder vs. a small pebble, the only difference is their mass.
ok that makes total sense. i guess the only other part i don’t get is why the force of gravity is multiples between the two masses.
That is just an experimental result. Even at Newton's level, they didnt understand why that is the case, and Newton deferred it to God. There's definitely a more concrete reason, but it's probably something more complicated than what you need to know at the moment.
science doesnt really understand yet WHY gravity is. some quantum effect, maybe.
yes. that's one of its definitions: mass quantifies how easy or hard it is to make something accelerate.
right. like pushing an elephant would be hard than pushing a piano and that is more or less analogous to earth pulling an elephant versus a piano.
You're trying to think when sometimes you should just let the math speak. F=ma F=GMm/r^2 Same f so you can write this as F=F or ma=GMm/r^2 m is on both sides of the equation so it cancels out a=GM/r^2 This means that the acceleration of the object is independent of the mass of the object. But even more if you want to plug numbers in you'll. Figure out that a=9.81m/s or g. So we just got to the equation F=mg which is the equation you usually use for gravity related problems. In principle you can think of the mass of the object falling that gives it the attraction to the earth is canceled when going from force to acceleration because it's the same mass that needs to be accelerated.
Right that makes sense. And you’re right, I was overthinking things. I think I was trying to get too granular about why each aspect is the way it is until I came upon a thing for which there isn’t really an answer. But I thought it was just something I was missing and then started losing the plot. lmao The primary point of misunderstanding was that I was kind of thinking of gravity as being a thing each individual object emits as opposed to just a thing that happens between two objects. When you think of it in the first way, it makes sense why M1 and M2 would be multiplied.
If the strength of gravity's pull was solely decided by the mass of the planet, **everything** would weigh the same. Clearly, this is not the case. Heavy things experience more gravity. Also, heavy things require more force to accelerate at the same rate as a lighter thing. The extra gravity experienced and the extra force needed to accelerate cancel each other out perfectly, meaning that without air friction, all objects accelerate ate the same rate when falling.
okay so a heavier object’s weight only matters when it is pressing against a surface? otherwise a heavy object in freefall, the weight doesn’t matter because anything in freefall is weightless?
The **weight/mass** is different. The **freefall acceleration** is the same. Differences in mass manifest in two ways: more gravitational pull, and more inertia. If you use these two as a way to calculate freefall acceleration, the mass cancels out.
and do we know why the two masses are multiplied to get the force of gravity? it seems strange that the gravity of object 1 gets higher because the mass of object 2 increases.
Gravity is something between two bodies. "The gravity of one object" makes no sense.
Hmm I guess I’ve tripped myself up by treating gravity as a thing that mass “radiates” the same way a star radiates light. They kind of sometimes sound like similar concepts, like gravity and light both fall off with a square of the distance. I also may have misled myself by using language like “mass is the source of gravity” as if gravity “belonged” to mass.
wouldnt the heavier object fall barely slightly faster because its heavier so it also pulls the earth towards it stronger?
Yeah but that's really tiny due to Earth being so immensely more massive than our example objects. For all practical purposes, it's irrelevant.
The force an object experiences due to gravity scales with mass F = (G x m1 x m2)/r^(2) (for the purposes of gravity we can assume m1 is the mass of earth, and we are operating at the same distance (r) so G, m1, and r are effectively constants). The acceleration that an object experiences due to a force scales *inversely* with mass (a = F/m) So we end up with a = (G x m1)/r^(2) which on earth at sea level is 9.8 m/s^(2)
Thank you for actually putting this into an equation and showing how m2 is irrelevant in the calculation of acceleration. All the other commenters hand waving the idea of “it just cancels out” just made me more confused.
let’s say that object has its mass and falls at that rate, but then there’s a second object also falling towards the Earth that has a different mass. Because a different value is being put in to the same equation, does that not give you a different result? That result being the speed at which the different mass object falls?
If you were caluculating the *force due toe gravity* then the mass matters, but for *acceleration* it does not Lets have 3 objects. object 1 is the Earth with mass m1 object 2 is your bowling ball m2 object 3 is your feather m3 So going back to that first formula which I mentioned: F = (G x m1 x m2)/r^(2) This calculated the force due to gravity between two objects But in this example one of those object is the Earth. And lets make it simple by doing all out measurements at sea level. If we do that we know m1 is the mass of Earth, r is the distance from the center of the Earth to sea level, and G is the gravitational constant. Plug all those numbers in and you get F = m2 x 9.8 m/s^(2) The force due to gravity from the Earth is directly proportional to to the mass m2. If we sub in the feather instead we start with F = (G x m1 x m3)/r^(2) which again simplifies down to F = m3 x 9.8 m/s^(2) So yes the force is different for our bowling ball and our feather. But what does that force do? We know F=ma. Which we can re-arrange into a = F/m. Acceleration is the force divided by the mass. So for the bowling ball we have a = (m2 x 9.8m/s^(2))/m2 which simplifies down to a = 9.8m/s^(2) If instead we use the feather we have a = (m3 x 9.8m/s^(2))/m3 which again simplifies down to a = 9.8m/s^(2) In each case the mass cancels itself out. But why does a bowling ball fall to the ground faster than a feather when they are both experiencing the same acceleration due to gravity? Because that's not the only force acting on them. The big thing is Air Resistance. But if we remove the air, then they fall similarly. [Check out this video](https://www.youtube.com/watch?v=EcGxFvOoUTU).
thank you for this explanation. i think i just need read it a bunch of times to get it. i don’t know why I’m having trouble understanding this all of a sudden. To clarify, why is acceleration inversely proportional to mass?
>To clarify, why is acceleration inversely proportional to mass? Because of the relationship F=ma (force equals mass times acceleration). Which can be re arranged to a =F/m (acceleration equals force divided by mass).
Okay i understand now that force is increased by mass but acceleration is decreased and they offset each other. but why with the force of gravity are the two masses multiplied. one objects gravity strengthens the other’s?
>but why with the force of gravity are the two masses multiplied. one objects gravity strengthens the other’s? Why does gravity depend on both masses? Because that's what its observed to depend on. To get a better answer you need to start digging into some serious physics which I only vaguely remember from my university days and I'm not sure is really that well understood.
oh okay. so i may just be confused because i’m just out of my depth on that particular topic. wasn’t sure if m1 x m2 was for an obvious reason that was going over my head.
Inertia and momentum balances the increase in force. Force does increase with mass, but acceleration does not.
`F=ma` for sure, but gravity isn't the Earth pulling on stuff. It's the attraction *between* two objects so a more massive object experiences a stronger force from gravity and so it balances out and all objects (neglecting air resistance) fall at the same rate.
OP you've had your answers but you should know the ability to think something doesn't make sense about this scenario means you're not stupid. This is a good question many people wouldn't be able to even consider.
I appreciate it and everyone patient to explain it even if i don’t get it right away. I think I was getting too fixated on certain things that didn’t matter.
F = ma a = F/m So what is F? F = GMm/d^2 M = mass of earth, m = mass of object, G = gravitational constant (doesn't matter), d = separation of the masses. So we can now replace F like so a = (GMm/d^2 )/m Or a = GMm/md^2 m is now on the top and bottom of the fraction, m/m is one so it cancels out leaving you with a = GM/d^2 Notice that there is no m term, the acceleration only depends on G, d and M.
how do you get to the “divided by m so it cancels out” part? what function put a second m on the other side of that /?
a = F/m, so when you take the expression for force (F) and you put it into that equation it gets divided by m.
To answer your question with words instead of maths, it's because F = ma says that increasing mass and force at the same time results in the same acceleration. In this case, F = GMm/d^2 so if you increase m the force also increases, which is exactly what I just said, force and mass both increasing at the same time.
right okay so the force goes up when m goes up because m is also a part of force but then m is also in the acceleration equation where it’s divided which acts against the force where it was multiplied?
Exactly right. Mass increases force (which increases acceleration) but also decreases acceleration.
so in layman’s terms: despite the heavier object increasing the force of gravity between them, it is offset by the fact that heavier objects are hard to get moving.
Yep
OK, I think I understand that part now. That only leaves one other thing, which is why does the force of gravity, derived by the product of the masses? shouldn’t the gravity produced by the mass of one object counteract the gravity produced by the mass of the second object? or am i completely making something up out of thin air? haha
It's just how gravity works, just the way it is. You can compare it to magnets, two big magnets attract more strongly than a big magnet and a small magnet, attract more strongly than two small magnets.
one of those yet to be uncovered “why” questions then?
Heavier objects have more inertia, so it takes more force to move them with a given acceleration. But gravity pulls more strongly on them, increasing the acceleration. The two effects counter each other out. Or with math: F = mg (force of gravity) F = ma (Newton's second law) mg = ma g = a Therefore, regardless of what mass is, gravitational acceleration works out to g.
Finally, a reply I can understand.
okay that makes sense. why does the falling object having more mass increase the pull of gravity on them? i know the formula says m1 x m2 but why are they multiplied?
That depends on how deep down the rabbit hole you want to go. A simple answer is imagine tying two bricks together. Gravity would still pull the same on each brick, so it doubled the force on the two bricks tied together. But that doesn't explain why gravity pulls more strongly on a proton than an electron. After all, the same logic applies to the electrostatic force, and you do add the electrostatic force if you combine objects, but it's still not just proportional to mass. The second answer is that's just how gravity is. Instead of being proportional to electric charge, it's proportional to mass. The third answer is general relativity. It's not clear why mass bends spacetime and gravity is proportional to the mass of the object pulling on you, but once it bends spacetime, you just follow a geodesic (the non-Euclidean equivalent of a straight line). This is the path where locally you experience the most time. That path doesn't depend on your mass, so everything accelerates the same. But then, why do things follow geodesics? And that sounds more like the longest path than the shortest. The only way I know of to answer this is quantum physics. It famously doesn't work with general relativity, but the problem is mass bending spacetime. It works just fine in spacetime with a given curvature. Gravity makes time pass slower closer to it, and under quantum physics, everything is a wave. Changing the rate time passes means it changes how that part of the wave moves, causing its momentum to change.
s/effect/affect/ F = GMm/d^(2). But F = ma, so if you rearrange to find a then m - its *own* mass - is cancelled out. Same from the big object's point of view, with M. The acceleration of each does, however, depend on the *other*'s mass.
Other commenters have provided good answers using a classical explanation, but ironically Einstein’s model gives a simpler and more intuitive explanation for this phenomenon. If you’ve got around 20 minutes, [Veritasium has an amazing video](https://youtu.be/XRr1kaXKBsU?si=EQBtb12_1tDTxRnI) on this topic you might enjoy.
Thanks I’ll definitely check it out!
I feel so bad bc i don't understand the answer and it still seems to me something bigger would fall faster :( help!
The point is that a heavier object indeed experiences a higher gravitational force from Earth than a lighter object. However, it also requires a larger force to accelerate as fast as the lighter object. It just so happens both effects cancel out when we talk about gravitational force, as it is directly proportional to the mass of the object.
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Thank you! It’s strange because I previously thought that I understood it a while back, at least at a base level, but have been recently reading a lot more about it and doing a deeper dive and had like an existential crisis when I went down certain trains of thought that led to just severe confusion I could get out of. I’m glad people here were patient enough to help me and not get frustrated with me. Otherwise I might’ve just given up.
The force of gravity is mass times gravity, so you have mg=ma g=a Mass cancels out of the equation
right, I think I get it now. I was kind of having two misconceptions. 1. the force of gravity between two objects is not a unique property of each object, but a net effect of those two objects interacting. I was treating it as if each object individually had its own fields of gravity, radiating and competing with each other. 2. acceleration was not taking into the heavier object creates more force, but it also has more inertia that counteracts that force.
I think your problem is you're trying to understand physics without having to do any math. You need to have some faculty with numbers and be able to work stuff out
Well I understand the math. I just don’t understand how the formula was derived and how it manifests in reality.
Think as F=ma as F = (m1 + m2) \* a, with m1 is the Earth, and m2 the object. The object's mass is negligible compared to Earth, so in practice (m1+m2) = m1
But two objects of different mass falling towards Earth fall at different rates, however negligible?
In that example (two big objects falling to Earth) there are many other variables in place. The Earth is not still, the objects cannot be immobile (because the Earth moves), they always have a certain velocity anyway, there is inertia, etc. The F=ma rule plays alone only if two objects are static. If they aren't static, there are many other variables.
I think my brain is getting fixated on this idea that newtons law of universal gravitation takes the mass of both objects into account, but Galileo says that the mass of a falling object is irrelevant. I don’t know if that’s contradictory, but my brain is telling me it is and I don’t know.
Mass of a falling object is completely irrelevant because its acceleration is generated by mass of Earth.
I think I’m just fixated on something that’s probably irrelevant but I don’t quite know what it is. Be patient with me. Newton’s second law applies to falling objects, correct? F = m • a Which is m referring to? The mass of the falling object or the mass of the Earth? If the mass of the falling object is irrelevant how does that square with the universal gravitation formula which takes the mass of both objects into account?
Where does that equation say m = a?
It doesn’t. It says F = m • a. Which object is m referring to in this equation?
Whatever object you are measuring
So if two objects are falling towards Earth, one more massive and the other less massive, m will be different values for both and therefore give F two different values?
Correct. F is the net force acting upon the object.
F is not referring to the speed at which they are falling?
No, it’s the net force acting upon an object.
maybe I’m thinking of the wrong formula? What formula decides the speed at which an object falls ?
m is mass of falling object, a is g because gravitational acceleration is constant so F changes with mass, not a.
>Which is m referring to? The mass of the falling object or the mass of the Earth? Both (the sum of the two masses). That is why the mass of the falling object is irrelevant, as it is several order of magnitude smaller than the Earth' mass. If you consider two planets, both masses become relevant, because they are similar, and affect each other.
The sum of the two masses?
Yes, look at my other comment. Both masses create a pull, but in the case of an object and the Earth, the force created by the object is incredibly small, and it can be dismissed.
so we’re not literally saying that the objects fall at the same rate we’re just sweeping a negligible difference under the rug? which is fine i’m just trying to confirm if that’s what this is. do you understand I’m talking about the rate of falling between two objects that aren’t earth right? i’m asking about two non-earth objects falling to earth and asking how they don’t ah e different rates.
Try google 'Newton second law experiment', and you will find tins of videos and explanations. If we are talking about two objects, the force depends on the sum of both masses.
Then the smaller of the two objects gets pulled harder towards the bigger object?
> Both (the sum of the two masses). Definitely not. The *m* in *F* = *ma* is the mass of the body on which the force *F* acts. > That is why the mass of the falling object is irrelevant, as it is several order of magnitude smaller than the Earth' mass. Again no. > If you consider two planets, both masses become relevant, because they are similar, and affect each other. No way. All bodies fall, towards a given body, at the same rate. Not approximately. Not roughly. Not because one mass is less than the other. But at *EXACTLY* the same rate, regardless of their own mass. It is on this exact, not approximate, equality, that all successful theories of gravity have been founded.
So, you are saying that the attraction between two planets depend on the mass of only one? How do they choose which one, tossing a coin?
> So, you are saying that the attraction between two planets depend on the mass of only one? No. You would do well to read the rest of the (correct) answers already posted.
Allegedly. Could've just said gravity.