Moment of Inertia and Rotations
Name: Tracy M.
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.
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
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,
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.
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
Illinois Central College
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Update: June 2012