Does inertia have anything to do with bodies falling at the same rate?

[Grasshopper]

The equation manipulation that shows that bodies will fall at the same rate regardless of their mass is very straight forward, because mass cancels when you set F = ma of the body equal to gravitational force using Newton's gravitational equation. I have no problem understanding that in terms of the extremely simple algebra.

However, looking at the equations, even if the accelerations due to gravity of a feather and comet in free fall are the same, their respective force equations are not, since for the feather, F_f = m_fa is much smaller than the force equation of the comet, F_c = m_ca.

So, I was wondering if the fact that the more massive object also has more inertia than the feather has anything to do with it. By that I mean, while the comet is significantly more massive, the force required to move it is also much larger, which would (I'm assuming) exactly counter whatever additional force that the much larger mass supposedly pulling on the Earth would contribute, resulting in the mass of the comet not mattering at all with respect to its acceleration as it falls.

I'm not in any way married to this explanation, by the way. I am simply interested in the truth, even if my intuition is way off, so by all means please help me find the actual explanation (after all, the point is to develop physics intuition, not become a crank ;) ).
Thanks to all who reply.

 

[Ibix]

Well, inertia is just a word for resistance to change of state of motion, which is what the $m$ in $F=ma$ does. So I think you are more or less stating the mathematical explanation, but substituting slightly less precise terms.

If you want an intuitive explanation, how about this one (from Galileo himself). Imagine that heavy objects do fall faster than light ones. Take a light ball and a heavy one and drop them and the heavy ball hits first. Now tie them together with a thread with plenty of slack and drop them again. Do they fall at different rates until the thread goes taut and then find some average rate? Or do they immediately drop even faster because they are one even heavier object? If you believe heavy objects fall faster than light ones it has to be the second one, but it's clearly silly.

 

[Grasshopper]

After I re-read my post I kind of assumed I was basically just restating the equation, but even though it's less precise, that and the other examples does clear out some incorrect intuition I had (I knew it was wrong but it would not go away).

I actually saw an old program that mentioned some of his thought experiments on that problem. I am not entirely sure if it was him, but I believe the program mentioned the notion of tying a feather to a hammer and dropping them. Does the feather cause the hammer to slow down? Or does the added mass make it fall faster? Or it was something to that effect.

 

[A.T.]

...but I believe the program mentioned the notion of tying a feather to a hammer and dropping them. Does the feather cause the hammer to slow down? Or does the added mass make it fall faster? Or it was something to that effect.

Maybe it was two masses, dropped side by side. If they accelerate just like when dropped separately, then fall acceleration is independent of mass, because we can consider the two masses side by side to be just one bigger mass.

 

[jbriggs444]

By that I mean, while the comet is significantly more massive, the force required to move it is also much larger, which would (I'm assuming) exactly counter whatever additional force that the much larger mass supposedly pulling on the Earth would contribute, resulting in the mass of the comet not mattering at all with respect to its acceleration as it falls.

Yes. This is correct.

In principle there are three quantities that might be called mass: inertia, active gravitational mass and passive gravitational mass.

Inertia is the m in $F=ma$.

Active gravitational mass is the M in $F=G\frac{mM}{r^2}$. The mass of the object that is doing the attracting.

Passive gravitational mass is the m in $F=G\frac{mM}{r^2}$. The mass of the object that is being attracted.

Newton's third law tells us that active and passive gravitational mass are proportional. You cannot change the active gravitational mass of an object without also changing its passive gravitational mass -- else the object could attract more strongly than it is attracted. Or vice versa.

The equivalance principle, the fact that all objects free fall at the same rate, tells us that inertia and passive gravitational mass are proportional. You cannot change the passive gravitational mass of an object without also changing its inertia else the fall rate would change.

There is no point having three separate units for quantities that are proportional to each other. So we use the same unit of measurement for all three and adopt the point of view that it is really just one quantity that is being measured.

 

[anorlunda]

some incorrect intuition I had

I like to think of Newton's 2nd law as $\frac{F}{m}=a$. That reminds me that the acceleration is proportional to the ratio of force to mass. More massive objects, more force, but the same ratio.

 

[hmmm27]

Gravity is only a "force" through convenience : there is no energy expenditure in its influence.

If you had a feather and a cannonball proximous in outer space (and had lots of time to spare), you'd notice that the light feather moves faster than the heavy cannonball, to their meeting.

Which looks like inertia, and could be explained as such if "gravity" were an attractive force emanating from a point equidistant between the two, but in reality there's two gravitational sources, and the cannonball is simply exerting more gravity on the feather than the converse.

(I may be stretching my understanding somewhat, but in that inertial frame, both objects have equal (and opposite) momenta as they move together towards their combined center of gravity, which position will remain constant, throughout.)

A cannonball sitting on a table in the lounge is exerting $f=ma$ on the table but gravity - while causing the force, in combination with the mass of the cannonball - is not the force, itself.

 

[jbriggs444]

Gravity is only a "force" through convenience : there is no energy expenditure in its influence.

It's called gravitational potential energy. It can be stored up or expended. Any child who has gone sledding has experienced this.

If you had a feather and a cannonball proximous in outer space (and had lots of time to spare), you'd notice that the light feather moves faster than the heavy cannonball, to their meeting.

Yes, the feather moves faster than the cannonball. Same force, different amounts of inertia. Different accelerations.

A cannonball sitting on a table in the lounge is exerting $f=ma$ on the table but gravity - while causing the force, in combination with the mass of the cannonball - is not the force, itself.

It's not $f=ma$. It's $\sum f = ma$. The downward force from gravity and the supporting force from the table sum to zero. The acceleration is zero. You can weigh the cannon ball by measuring the deflection of the table and hence the supporting force it is providing. Hooke's law.

I think you are saying that "gravity" is the phenomenon and that the "force of gravity" is the force. Yes, that is true.

 

[hmmm27]

It's called gravitational potential energy. It can be stored up or expended. Any child who has gone sledding has experienced this.

Force is not energy. PE is a system property. Unlike a rocket, where the force is achieved by accelerating bits and pieces out the back at high speed, the Earth did not expend any energy to achieve its "force".

Yes, the feather moves faster than the cannonball. Same force, different amounts of inertia. Different accelerations.

Umm, no. Two sources of gravity, not one.

Inertia isn't the issue : if the feather were a second cannonball, it would gravitate (literally, hah) towards the first no slower than the feather did.

It's not f=ma. It's ∑f=ma. The downward force from gravity and the supporting force from the table sum to zero.

The (downwards) force is exerted by the cannonball, not gravity.

The acceleration is zero. You can weigh the cannon ball by measuring the deflection of the table and hence the supporting force it is providing. Hooke's law.

What happened to your ∑f=ma ? Unless you're proposing that the forces are equal because they're both zero.

I think you are saying that "gravity" is the phenomenon and that the "force of gravity" is the force. Yes, that is true.

Not really. It might be interesting to say "force imparted by gravity" instead of "force of gravity". Wouldn't change the equations any, of course.

 

[jbriggs444]

Force is not energy.

Yes. We agree about that.

PE is a system property. Unlike a rocket, where the force is achieved by accelerating bits and pieces out the back at high speed, the Earth did not expend any energy to achieve its "force".

Why this idea that force requires an expenditure of energy? Construction workers can lean on their shovels all day without expending energy.

[Feather and cannonball attracting one another]
Umm, no. Two sources of gravity, not one.

Yes. Both sources resulting in the same magnitude of force on the other object.

Inertia isn't the issue : if the feather were a second cannonball, it would gravitate (literally, hah) towards the first no slower than the feather did.

Yes. And it would be subject to a larger force. More inertia. Larger force. Same acceleration.

The (downwards) force is exerted by the cannonball, not gravity.

The downward force on the cannonball is exerted by gravity from the Earth, not by the cannonball.

What happened to your ∑f=ma ? Unless you're proposing that the forces are equal because they're both zero.

Upward force from table on cannonball. Downward force from gravity on cannonball. Sum of forces is zero. Acceleration is zero. $\sum f = ma$.

Not really. It might be interesting to say "force imparted by gravity" instead of "force of gravity". Wouldn't change the equations any, of course.

Exactly what are you disagreeing with here?!

 

[gmax137]

in reality there's two gravitational sources, and the cannonball is simply exerting more gravity on the feather than the converse.

What does this (the bold words) mean? The force on the feather is equal to the force on the cannonball, they are a Newton's Third pair, no?

 

[A.T.]

... the cannonball is simply exerting more gravity on the feather than the converse.

This is not how Newtonian gravity works. The gravitational forces on both objects have the same magnitude, which is proportional to the product of their masses.

 

[gmax137]

What do you mean when you say, "exerting more gravity"? These words are, AFAIK, not standard, and I have no idea what you are trying to say.

 

[hmmm27]

Why this idea that force requires an expenditure of energy? Construction workers can lean on their shovels all day without expending energy.

And the phenomenon responsible for that effortless lean is called ? Begins with a 'g'.

Yes. And it would be subject to a larger force. More inertia. Larger force. Same acceleration.

Inertia (named after a large Roman centurion 'Inertius' who could not be moved off his barstool) has nothing to do with it : it's the fact that the gravitational influence of the lighter body is less than that of the other.

The original post seemed (to me) in regards an intuitive misobservation that inertia is responsible for the massier body accelerating slower in a 2-body gravitational system... to be more precise, comparing 2 disparately massed bodies under the gravitational influence of a very massive body : why isn't the heavier of the two deflected less ?

The downward force on the cannonball is exerted by gravity from the Earth, not by the cannonball.

No, the force is a property of the cannonball - its mass and its acceleration - not the Earth, which only imbued(induced? sorry, English is my first language) the acceleration.

Upward force from table on cannonball. Downward force from gravity on cannonball. Sum of forces is zero. Acceleration is zero. ∑f=ma.

sp. "Downward force from cannonball on table".

Exactly what are you disagreeing with here?!

Your assertion that we agreed on something, if I recall correctly.

What does this (the bold words) mean? The force on the feather is equal to the force on the cannonball, they are a Newton's Third pair, no?

Acceleration is being imbued, not force.

This is not how Newtonian gravity works. The gravitational forces on both objects have the same magnitude, which is proportional to the product of their masses.

Thanks, forgot about that part.

What do you mean when you say, "exerting more gravity"? These words are, AFAIK, not standard, and I have no idea what you are trying to say.

Is more gravid ? Its gravitational pull is greater.

 

[jbriggs444]

And the phenomenon responsible for that effortless lean is called ? Begins with a 'g'.

The force between man and shovel is not gravity. Try again. [Hint: it is a contact force]

Inertia (named after a large Roman centurion 'Inertius' who could not be moved off his barstool) has nothing to do with it : it's the fact that the gravitational influence of the lighter body is less than that of the other.

And because the gravitational force is larger but the observed acceleration is the same, we conclude that the inertia is larger as well.

Your etymology for inertia is suspect, btw.

The original post seemed (to me) in regards an intuitive misobservation that inertia is responsible for the massier body accelerating slower in a 2-body gravitational system... to be more precise, comparing 2 disparately massed bodies under the gravitational influence of a very massive body : why isn't the heavier of the two deflected less ?

That is an interesting take. Let me go back and review the OP. Nope. That interpretation does not stand up. It's "why isn't the heavier of the two deflected more".

No, the force is a property of the cannonball - its mass and its acceleration - not the Earth, which only imbued(induced? sorry, English is my first language) the acceleration.

No.

That is not how Newtonian forces work. You have a force by one object on another. There is a force on the cannonball by the earth.

The direction and magnitude of that force is given by Newton's universal law of gravity.