This month's Challenge Question: Specs & Techs from GlobalSpec:
Two
cylinders of equal length and radius are set at the same height on a ramp and
allowed to roll to the bottom. The first is a solid aluminum cylinder. The
second is a hollow lead cylinder with an inner radius slightly more than 3/4 of
its outer radius. Assuming the frictional effects are negligible, which
cylinder reaches the bottom of the ramp first?
And the answer is:
The solid aluminum cylinder will reach the bottom first.
The key to this problem is that the cylinders are rolling
down the ramp. In this case the masses of the cylinders are unimportant, for the same reason that they would be unimportant if the cylinders were just
dropped from the same height (neglecting air friction). What matters in this
problem is the difference in the moment of inertia of a solid cylinder as
compared to a hollow cylinder (cylindrical shell). This difference leads to
different accelerations for the respective cylinder's center of mass.
Center of Mass Acceleration of the solid cylinder:
Where g is the acceleration due to gravity (9.8 m/s^2) and θ
is the angle of the ramp.
Center of Mass Acceleration of the hollow cylinder:
Where g is the acceleration due to gravity (9.8 m/s^2) and θ
is the angle of the ramp.
Since the acceleration of the center of mass of the solid
cylinder is greater than the acceleration of the center of mass of the hollow
cylinder, the solid cylinder reaches the bottom of the ramp first.
