Galileo's Expements

[alan2here]

I'm making sure I understand this.

 

Galileo dropped a piece of lead shot and a lead cannonball off the top of the tower. Because there where the same density they hit the ground at the same time irrespective of them being different weights and sizes (they where both made of lead therefore both the same density). However if it had been a plastic ball and a lead ball of the same size they would have been different densities as well as different weights and the lead one would have hit the ground first?

[insane_alien]

in a vacuum, yes but with air resistance, no they would not have hit at the same time.

[alan2here]

What did Galileo prove then? I know in a vacume it is even more extreme, in a vacume the plastic ball and the lead ball of the same size will it the ground at the same time despite the difference in density. Did he ignore air resistance because in his example the effect is so small?

[insane_alien]

well, in his experiment the ability to make the measurements depended mostly on the human eye. air resistance was obviously small enough to make it appear simultaneous. especially since the balls were released by humans as well.

 

of all the factors that introduced error, air resistance was one of the lesser.

[timo]

Galileo dropped a piece of lead shot and a lead cannonball off the top of the tower.

I think the part of dropping stuff from the tower of Pisa is just a myth. Doesn't matter for the physics, though.

 

Because there where the same density they hit the ground at the same time irrespective of them being different weights and sizes (they where both made of lead therefore both the same density).

The common view is that density does not play a role, either - you could as well do the thing with lead and gold (I propose gold nuggets rather than gold cannonballs for economic reasons).

What the experiment is supposed to show is that when you can sensibly neglect air resistance, the masses of the objects do not matter for the time it takes them to hit the ground. More specifically, the mass-terms in the gravitational force acting on an object (heavy mass) [math] F_G = G \frac{m M_0}{r^2}[/math] and the mass term resisting acceleration (inert mass) in the equation of motion [math] F=ma [/math] cancel: [math] ma=F_G=F=G \frac{m M_0}{r^2} \Rightarrow a = G \frac{M_0}{r^2}[/math]. This does not hold true when [math] F \neq F_G[/math] like when a non-neglectible friction force works on the object (i.e. [math] F = F_G + F_F [/math] with [math]F_F[/math] being friction-force).

 

However if it had been a plastic ball and a lead ball of the same size they would have been different densities as well as different weights and the lead one would have hit the ground first?

.

The plastic ball would hit the ground later than the cannonball, but that's because it's affected by friction stronger than the cannonball. Similar experiments as the original one have been repeated often in the past dunnohowmany-hundred years. The most interesting ones imho are the ones performed under conditions where there is not friction, that is in vacuum conditions. It's a relatively widepsread school experiment, actually: Take a glass cylinder with a feather and a stone inside. Pump out the air and then turn the cylinder upside down -> both objects will fall at the same speed as above calculation suggests. Same thing has also been performed as a fun experiment on the moon; you might even find the video of it somewhere on the net, but the video quality is quite bad.

 

EDIT: Considering the follow-up question:

The air resistance of a leaden cannonball is ... errr ... more negligible than the effect of air resistance on a plastic ball. Like I said the thing about throwing the balls from a tower is perhaps just a myth. It seems more practical to roll the balls down a ramp to keep the velocities (air resistance increases with velocity) in a reasonable range.

[alan2here]

So density plays a role in air resistance but not gravitationally in how fast the objects fall. Therefore the lead shot and cannonball will land at almost the same time with just a bit of difference due to air resistance but the plastic and lead balls of the same size but differant densities will land at very different times because of air resistance.

[timo]

Density does not directly affect air resistance but it does indirectly. Frictional force is a function of the object's geometry (=shape) and velocity - physicists sometimes write this in a shorthand notation [math] F_F = F_F(\text{geometry},v) [/math]. Now let's assume some geometry and velocity are given and look how mass will affect the acceleration: [math] ma = F = F_G + F_F(\text{geo},v) = G \frac{m M_0}{r^2} + F_F(\text{geo},v) \Rightarrow a = \underbrace{G \frac{M_0}{r^2}}_{a_{\text{free}}} + \underbrace{ \frac{F_F(\text{geo},v)}{m} }_{a_\text{frict}} [/math]. The term [math]a_\text{free}[/math] is independent on the mass of the object. The term [math]a_\text{frict}[/math] (which has a different sign than [math]a_\text{free}[/math], meaning it's magnitude has to be substracted from [math]a_\text{free}[/math]'s magnitude - I just dropped the vectors in favour of readabillity) however depends on the mass. Therefore due to the extra term [math]a_\text{frict}[/math] increasing mass will lead to increasing acceleration once you consider friction.

Now to come back to how density affects acceleration: Keeping geometry the same and considering the same current velocity, higher density will lead to a greater mass, decrease the magnitude of [math]a_\text{frict}[/math] and increase acceleration a.

 

EDIT: Or in short and trying not to overcomplicate things: Yes, that (your previous post) is more or less true.

[waitforufo]

Okay, I think some of you are making this a little too complicated. We are talking about Galileo here. Galileo had a hypothesis that earth’s gravity caused all objects to accelerate towards it at the same increasing velocity if friction could be ignored or equalized. Galileo equalized the force of friction by making balls of the same size from different materials. The balls therefore had different masses. Galileo performed experiments to prove his hypothesis correct by rolling balls down a precision declining plane. Along the plain he suspended bells that would ring when struck by a rolling ball. He could move the location of the bells so that the bells rang with equal timing when compared to a swinging pendulum or metronome. By measuring the successive increases in distance between the bells, he could determine the change in velocity of a particular ball for a set decline. The balls all had the same velocity change regardless of their mass (or material of construction.) In other words he did not have to change the positions of the bells after setting them up with the first ball.

 

Galileo was known to be quite a showman. He may have indeed performed a demonstration of his discovery by dropping balls from the tower of Pisa, but this story is likely a myth.

 

An interesting side note of history is that Galileo was not too impressed with this discovery he made early in his life. He had no intention of publishing the results. Galileo was know to enjoy his public notoriety and was concerned that while under arrest by the Church he would slip into obscurity. To keep this from happening he searched his early notes to find topics that the Church would allow him to publish. The Church found nothing offensive in his silly rolling ball experiments and allowed him to publish. Newton, when reading these experiments, did not consider them so trivial.

[Country Boy]

First, let me point out that the " Because there where the same density they hit the ground at the same time" is not true. The density is irrelevant.

 

Also what Galileo really did is roll object down inclines. That way he could control the acceleration and so speed so he could watch more carefully. Of course, he found that both light and heavy objects rolled down the incline in the same time. As long as you are using objects that are fairly heavy for their size (iron, small wood spheres, etc.) air restistance doesn't mess it up too badly. What he as really trying to do is disprove Aristotle's claim that the acceleration would be in PROPORTION to the mass so small differences in time were not a greate problem.

[waitforufo]

If you want to see an example of the experiment go to

 

<http://brunelleschi.imss.fi.it/museum/esim.asp?c=404013>