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Name: Cecilia
Status: other
Grade: 4-5
Location: NJ
Country: N/A
Date: 2/1/2006


Question:
My daugther did an experiment with 2 exact size and shape water bottle. She rolled then down two same size planks and the faster one was the water bottle. Everything we read says the lighter bottle should be faster but its not. Can you help her explain this.


Replies:
The way you asked the question was a little unclear. But I'll venture a guess that a big, thin-walled, empty bottle (full of air) would be slowed down a few percent by air-drag if it rolled to a level 3 or more feet lower. The same bottle filled with water would be heavier but have the same air-drag, so it would be slowed down less than 1% by air-drag, and would be a few percent faster than the empty bottle.

Heavy objects falling at the same rate as lighter objects is usually not quite true in air. In vacuum with no air-drag the truth of it becomes clear.

There's another reason, too. When rolling down a ramp, the angular inertia adds to the ordinary inertia. Just like it takes effort to accelerate a big mass, it also takes effort to start it rotating. An object with all it's mass around the rim of the rotation has more angular inertia than one with all it's mass in the center. In between is the object with mass uniformly distributed throughout it's volume The plastic bottle filled with water or ice is an example of uniform-distribution. the empty plastic bottle may have more of it's weight around the rim (especially if you cut off the thick, heavy top and bottom of the bottle).

Suppose you had three bottles equal not only in size, but also in mass & weight. Bottle#1 was filled with melted wax and allowed to cool and harden. Bottle#2 had lead BB's glued all over the inside wall, the right amount to weigh the same as #1. Bottle#3 had the same number of lead BB's in a clump supended in the center of the bottle. (how I dunno, maybe with strings & more glue) prediction: Bottle#2 would be slowest, because all it's mass is at the rim, maximizing it's angular inertia. Bottle#3 would be fastest, because with all the mass in a small center-point, it would have little angular inertia. Bottle#1 would finish between #3 & #2, because some is in the center, and some is near the rim.

The third concept is that the water is liquid, so it doesn't all need to rotate immediately. Sure it's rotating by the time the bottle reaches the bottom, but maybe not as fast as the bottle itself. And at the very start, the bottle can roll a couple of turns before the water in the center gets rotating much. This can make the water-filled bottle slightly faster than an ice-filled bottle, I suspect.

If there is serious bouncing or rolling drag, that might matter, too.

Why, in your ideas, was the lighter bottle supposed to be faster? True, it has less inertia, but it is also pulled less by gravity, and for falling straight down in a vaccum, these changes are supposed to exactly match. For rolling down a ramp, the only new thing is rotation. Otherwise the inertia matches the gravity force. If you want to have a change in inertia but have the same force, you need to make the force with a mechanical spring, not with gravity. I'm having difficulty finding any reason that might tend to make the ligher bottle faster. Three reasons would make the heavier bottle faster: air-drag, water's liquidity, and the angular-inertia difference due to different mass distribution.

Maybe you should roll them down the ramp on small & equal wheels (roller-skates),

James Swenson


I'm curious about a few other aspects of this expieriment. Were both bottles round? was this done inside, away from the wind?

Classicly, both bottles should roll down a ramp at the same speed, as the force required to accellerate them is derived from thier own weight. In practice, the bottle of watter probrably accellerated and rolled down faster because it's mass rendered other resistances trivial, while the empty bottle had to 'work' harder to overcome everything from air resistance to friction.

Ryan B.



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