In a pendulum mill, shown schematically in Fig. P16.6, the grinding is by a conical muller that is free to spin about a pendulous axle that, in turn, is connected to a powered vertical shaft by a Hooke universal joint. The muller presses against the inner wall of a heavy steel pan, and it rolls around the inside of the pan without slipping. The weight of the muller is W = 980 lb; its principal mass moments of inertia are Is = 121 in·lb·s2 and I = 88 in · lb ·s2. The length of the muller axle is l = RGA = 40 in, and the radius of the muller at its center of mass is RGB = 10 in. Assuming that the vertical shaft is to be inclined at θ = 30◦ and will be driven at a constant angular velocity of ωp =

240 rev/min, find the crushing force between the muller and the pan. Also determine the minimum angular velocity ωp required to ensure contact between the muller and the pan.

Figure P16.6