A sphere of mass m and radius R is welded to a massless rod as shown...

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A sphere of mass m and radius R is welded to a massless rod as shown (the sphere cannot spin relative to axis AB). The rod is attached by means of a frictionless pin at point A to a vertical shaft that precesses at rate . At low (constant) rotation rates the sphere rests against the vertical shaft making contact at point C. But at some critical speed, = crit, the normal force at C goes to zero. For rotation rates > crit, the sphere is free to oscillate with time varying angle 0(t). The distance from A to the mass center of the sphere, B, is L = 3R. Recall that the mass moment of inertia of a sphere is (2/5)mR². (a) Find the equation which, when solved will yield the critical rotation rate, crit. Assume that is constant. (b) Assuming that > crit, find the equation of motion for the angle 0(t). Use a Newton- Euler approach. $c A Z C less than crit Z B X A Z Z greater than crit 9 X A sphere of mass m and radius R is welded to a massless rod as shown (the sphere cannot spin relative to axis AB). The rod is attached by means of a frictionless pin at point A to a vertical shaft that precesses at rate . At low (constant) rotation rates the sphere rests against the vertical shaft making contact at point C. But at some critical speed, = crit, the normal force at C goes to zero. For rotation rates > crit, the sphere is free to oscillate with time varying angle 0(t). The distance from A to the mass center of the sphere, B, is L = 3R. Recall that the mass moment of inertia of a sphere is (2/5)mR². (a) Find the equation which, when solved will yield the critical rotation rate, crit. Assume that is constant. (b) Assuming that > crit, find the equation of motion for the angle 0(t). Use a Newton- Euler approach. $c A Z C less than crit Z B X A Z Z greater than crit 9 X

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