In Exercise 45, the case is made that the position uncertainty for a typical macroscopic o...

In Exercise 45, the case is made that the position uncertainty for a typical macroscopic object is generally so much smaller than its actual physical dimensions that applying the uncertainty principle would be absurd. Here we gain some idea of how small an object would have to be before quantum mechanics might rear its head. The density of aluminum, 2.7 X 103 kg/m3, is typical of solids and liquids around us. Suppose we could narrow down the velocity of an aluminum sphere to within an uncertainty of 1 µm per decade. How small would it have to be for its position uncertainty to be at least as large as  % of its radius?