Question 3.3.3: Four smooth spheres of equal radius form a pyramid on a smoo...
Four smooth spheres of equal radius form a pyramid on a smooth table (Fig. 3.3-3(a)). The bottom three spheres are constrained by a cylinder of such diameter that these three spheres are just touching each other and the cylinder. If each sphere weighs W, determine the contact force between each bottom sphere and the cylinder.
The contact forces H between the bottom spheres and the cylinder are due to the weight W of the top sphere, which is transmitted from the centre A of the top sphere through the centres B, C, and D of the bottom spheres to the cylinder, along the paths shown in Fig. 3.3-3(b). Fig. 3.3-3(c) shows a three-dimensional view of the regular tetrahedron ABCD together with the system of forces that act through the four vertices. For vertical equilibrium of the system, the vertical reaction V of the table acting through the centre of each bottom sphere must be
V=1\frac{1}{3} W
Hence, a triangle of force for point B, say, must be as shown in Fig. 3.3-3(d), from which
H = \frac{W}{3}\tan θ
Since the tetrahedron ABCD is regular, i.e. the edges are equal in length, it can readily be shown that tan θ = 1/√2. Thus
H = \frac{W}{3√2} =0.236W
