Why don't objects with greater mass fall faster?

[tarekatpf]

Galileo found in his experiment that all objects regardless of its mass take same time to fall from above, if the air resistance is effectively overcome.

Aren't objects with larger mass supposed to fall quickly, since objects with larger mass attract the Earth more strongly than smaller ones, and so there is stronger gravitational force between Earth and objects with larger mass than between Earth and objects with smaller mass?

I must be missing something.

 

[Dale]

Objects with a larger mass do have a larger gravitational force, but they also have more inertia meaning that it takes more force to accelerate them the same amount. Those two effects cancel out exactly.

In general, for any force which is proportional to the mass of the object on which the force is acting, the acceleration is independent of the mass:

f=km (for any force proportional to the mass)
f=ma (Newton's second)
km=ma (transitivity)
k=a (division by m)

 

[rcgldr]

Relative to the center of mass between two objects, the rate of acceleration of each object is determined by the amount of mass in the other object and the distance^2 between the center of mass of each object.

Labeling the objects as 1 and 2, then relative to the center of mass of the two objects, and defining the acceleration of object 2 as positive, then

a_1 = - G m2 / r^2
a_2 = + G m1 / r^2

The combined acceleration of the objects towards each other is a function of the total mass of the two objects:

a_combined = a_1 - a_2 = -G (m1 + m2) / r^2

 

[A.T.]

I must be missing something.

Drop two identical objects side by side. Why should they fall any faster, if you connect them somehow and consider it one large object, with double the mass of a single one?

 

[Khashishi]

Equivalence principle. According to Einstein, falling under gravity is "equivalent" to having the ground accelerate up toward you. If you have two objects in free fall, near each other, it's as if the ground were accelerating up toward them. The ground will hit them at the same time, regardless of mass.

 

[sophiecentaur]

Galileo found in his experiment that all objects regardless of its mass take same time to fall from above, if the air resistance is effectively overcome.

Aren't objects with larger mass supposed to fall quickly, since objects with larger mass attract the Earth more strongly than smaller ones, and so there is stronger gravitational force between Earth and objects with larger mass than between Earth and objects with smaller mass?

I must be missing something.

Back to intuition:
Jump out of a window, holding hands with someone. Ignore any effects due to the air. Would you expect your rate of fall to change if you suddenly let go hands? If your (correct) answer is "no" then you will see how an object of M (you, on your own) will fall at the same rate as an object of 2M (you and your mate, joined together).

 

[A.T.]

Drop two identical objects side by side...

Jump out of a window, holding hands with someone...

Yeah tarekatpf, you have the choice which of the two experiments you want to perform.

 

[sophiecentaur]

In my experiment, you'd be putting your money where your mouth is. A bit of personal involvement with an experiment is good for impact (haha).
You could always jump into water. . . . .

 

[Dale]

Yeah tarekatpf, you have the choice which of the two experiments you want to perform.

 

[tarekatpf]

Objects with a larger mass do have a larger gravitational force, but they also have more inertia meaning that it takes more force to accelerate them the same amount. Those two effects cancel out exactly.

In general, for any force which is proportional to the mass of the object on which the force is acting, the acceleration is independent of the mass:

f=km (for any force proportional to the mass)
f=ma (Newton's second)
km=ma (transitivity)
k=a (division by m)

Thank you.

 

[tarekatpf]

Galileo found in his experiment that all objects regardless of its mass take same time to fall from above, if the air resistance is effectively overcome.

Aren't objects with larger mass supposed to fall quickly, since objects with larger mass attract the Earth more strongly than smaller ones, and so there is stronger gravitational force between Earth and objects with larger mass than between Earth and objects with smaller mass?

I must be missing something.

Thank you very much.

 

[tarekatpf]

Back to intuition:
Jump out of a window, holding hands with someone. Ignore any effects due to the air. Would you expect your rate of fall to change if you suddenly let go hands? If your (correct) answer is "no" then you will see how an object of M (you, on your own) will fall at the same rate as an object of 2M (you and your mate, joined together).

Yeah, I jumped holding the hands of an octopus. I tried to get rid of it, but it didn't let me. Then suddenly a big bird came flying towards us, and let me sit on its back and then it ate up the octopus.

 

[tarekatpf]

Relative to the center of mass between two objects, the rate of acceleration of each object is determined by the amount of mass in the other object and the distance^2 between the center of mass of each object.

Labeling the objects as 1 and 2, then relative to the center of mass of the two objects, and defining the acceleration of object 2 as positive, then

a_1 = - G m2 / r^2
a_2 = + G m1 / r^2

The combined acceleration of the objects towards each other is a function of the total mass of the two objects:

a_combined = a_1 - a_2 = -G (m1 + m2) / r^2

Though I'm not good at getting those mathematical explanation, but thank you very much anyway.

 

[nix00]

That's the principle of equivalence from Galileo. You can see if you write the Newton's equation that the acceleration of an object in a gravitational field is independent of its mass, roughly speaking, in the vacuum, on Earth of mass M, with m the mass of any objects
$$F = m a =\mathcal{G} \frac{m M}{r^2} \rightarrow a =\mathcal{G} \frac{ M}{r^2}$$
and two differents objects of masses m1 and m2 will fall in the same way.

 

[sophiecentaur]

Yeah, I jumped holding the hands of an octopus. I tried to get rid of it, but it didn't let me. Then suddenly a big bird came flying towards us, and let me sit on its back and then it ate up the octopus.

What were you smoking at the time?

 

[sophiecentaur]

Though I'm not good at getting those mathematical explanation, but thank you very much anyway.

The trouble is that some level of knowledge of Maths is pretty essential, even for much of elementary Physics. Being without Maths is a bit like trying to play Chess when you don't know how the pieces are supposed to move or going to a foreign film with no subtitles and you don't know the language. You just have to believe the person you're with.

 

[OmCheeto]

I was going to respond in the negative, stating that the OP might have a point, and was going to do an experiment with balloons inflated to similar dimensions and filled with varying amounts of rocks.

But I don't have any balloons handy, and seemed to remember that someone did a similar experiment a while back, in the ideal location:

https://www.youtube.com/watch?v=5C5_dOEyAfk​

I'm fairly certain that this experiment would yield different results in my living room.

 

[A.T.]

A bit of personal involvement with an experiment is good for impact (haha).

But humans are notoriously unrelaible test objects. For example: If I was falling from a building holding your hand, I would instinctively pull you under me. And then jump off you just before impact.

 

[sophiecentaur]

Or I'd be doing that to you. (After writing copious experimental notes on the way down.)

 

[Drakkith]

Or I'd be doing that to you. (After writing copious experimental notes on the way down.)

You have far more self control in the face of your death than I would, good sir.

 

[sophiecentaur]

You are clearly not dedicated enough to your Physics, my friend.

 

[Drakkith]

You are clearly not dedicated enough to your Physics, my friend.

That's fair.