Question

A heavy spherical ball of weight W rests in a V shaped trough whose sides are inclined at angles α and β to the horizontal. Find the pressure on each side of the trough. If a second ball of equal weight be placed on the side of inclination α, so as to rest above the first, find the pressure of the lower ball on the side of inclination β.

Answer 1

 Let R₁ = Reaction of the inclined plane AB 

on the sphere or required pressure on AB 

R₂ = Reaction of the inclined plane AC on the sphere or required pressure on AC 

The point O is in equilibrium under the action of the following three forces: W, R₁, R_2.

Case – 1:

Apply lami's theorem at point O

R₁/sinβ= R_2/sin(180 – α) = W/sin(α + β)

or R₁ = W sinβ/sin(α + β)

and R₂ = W sinα/sin(α + β)

Case – 2: Let

R₃ = Reaction of the inclined plane AC on the bottom sphere or required pressure on AC Since the two spheres are equal, the center line O₁O₂ is parallel to the plane AB. When the two spheres are considered as a single unit, the action and reaction between them at the point of contact cancel each other. Considering equilibrium of two spheres taken together and resolving the forces along the Line O₁O₂, we get.

R₃ cos{90° – (α + β)} = W sinα + W sinα

R₃ sin(α + β) = 2 W sinα

Or, R₃ = 2W sinα/sin(α + β)