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A possible use for the Tower of Pisa

A contemporary of Galileo dropped simultaneously two hollow brass balls from the top of the leaning tower of Pisa. Both balls had a diameter of 100 mm and a weight of 2.0 N but the first ball was highly polished and smooth, while the second one was cast crudely and retained substantial roughness. There was no wind at the time of the experiment. The density of air is 1.2 kg/m3 and its kinematic viscosity is 15×10-6 m2/s. The gravitational acceleration is 9.81 m/s2and the volume of a sphere is πD3/6, where D is its diameter. The tower can be considered as sufficiently high for the balls to reach their terminal velocity.
a) Compute the speed of the smooth ball when it reached the ground.
b) Explain in detail which ball would have reached the ground first.
c) If the smooth ball were given a clockwise spin, which it maintained throughout the fall, sketch its likely trajectory. Explain the rational for this sketch.


Contributed by Stavros Tavoularis, Department of Mechanical Engineering, University of Ottawa, Ottawa, Canada.

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