A small bock with mass m is placed inside an inverted cone t

A small bock with mass m is placed inside an inverted cone that is rotating about a vertical axis such that the time for one revolution of the cone is T (Fig. 5.83). The walls of the cone make an angle beta with the vertical. The coefficient of static friction between the block and the cone is mu_s. If the block is to remain at a constant height h above the apex of the cone, what are the maximum and minimum values of T?

Solution

To find the normal force (cone \"wall\" pushing on block), we\'ll need to find components of weight and of the centrifugal force from the spinning cone.
N = mg sin + m²r cos

= 2/T and r = h tan, so
N = mg sin + m(2/T)²(h tan)cos
N = mg sin + m(2/T)²(h sin)

All of the other forces mentioned will occur in directions parallel to the surface of the cone.
For minimum T,
component of weight = friction + component of centrifugal force
mg cos = N + m²r sin
mg cos = [mg sin + m(2/T)²(h sin)] + m(2/T)²(h tan) sin
g(cos - sin) = h(2/T)²(sin)( + tan)
T = 2 sqrt{h(sin)( + tan) / [g(cos - sin)]}
T = 2 sqrt{h(tan)( + tan) / [g(1 - tan)]}

For maximum P,
component of weight + friction = component of centrifugal force
Since the only difference is friction changing direction, change the signs of the terms with :
T = 2 sqrt{h(tan)(- + tan) / [g(1 + tan)]}