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Name: Tracy M.
Status: educator
Age: 40s
Location: N/A
Country: N/A
Date: 3/24/2004


Question:
My son built a K'Nex ball factory. He did an experiment by filling one ball with sand, the other being empty. He put the balls down three different paths, each time the ball with sand reaching the bottom at a faster time. What Mom here remembers from physics is that objects dropped will fall at the same rate if there is no wind, etc. I cannot explain simply and correctly why the heavier object goes faster through the paths though I understand the gist of it. Could you please help? Thank you.


Replies:
If you just drop the balls, you have essentially the same physics, and it is a little less complicated to explain. The total force on the heavy ball is

Mg - D

where M is the mass, g is the gravitational acceleration, and D is the drag force (i.e., air resistance). Similarly, the force on the light ball is

mg - D

where m is its mass, and for the moment we will not get caught up in the detailed behavior of the drag force, D. We know it will depend on the size of the balls, and vary with their speeds, but let us just go with "D" for now.

The important thing to look at is the net acceleration. For the heavy ball, it is

A = (Mg - D) / M = g - D/M

For the light ball, it is

a = (mg - D) / m = g - D/m

So, the difference between the two balls is that, for any drag force D, the heavy ball is affected less -- D/M is less than D/m, because M is greater than m. As M gets very large, D/M starts to look like zero, and the ball is essentially unaffected by drag.

As m gets very small, our equation looks like it might be trying to say something stupid, like D/m might get greater than g, and the ball will be accelerated upward. But D depends on speed, and in fact goes to zero as speed goes to zero, so we do not actually have a problem.

Tim Mooney


Tracy,

The sand ball is not going faster because of its weight. Both balls have to move just as much, and gravity does this the same for both. Both balls also have to spin. This is the problem. All of the empty ball's material is at the outside surface. The whole ball has to spin a lot. The sand ball has a great deal of its material toward the center. The sand near the center does not have to spin very much. Try making a long rod spin around its center (that the ends of the rod both moving fast) and then around the axis (the ends of the rod not moving at all). The weight of the ball can make the sand spin quickly because not all the sand has to move much during the spin. The weight of the empty ball has to make much more of the empty ball move fast during the spin.

Dr. Ken Mellendorf
Physics Professor
Illinois Central College



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