Help with Past Exam Question on Rolling Disk on Hemisphere
Hoping to outdo his physics professor, Doofus wants to dramatically demonstrate parabolic motion by throwing a cheese wheel of mass m and radius r off of the top of the UW observatory, which is at a height 3R above the roof of the physics building, as shown in the diagram. Just as he is finishing his climb, he accidentally lets go of the cheese wheel at the top of the hemispherical dome of radius R. It begins rolling without slipping from rest down the edge of the dome.
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a) Draw a freebody diagram of the cheese when it has rolled a distance s along the dome
b) What is the value of s when the cheese leaves the dome
c) Diligent is standing on the roof of the physics building and catches the cheese just as it hits the ground. How fast is the cheese moving when he catches it
2. Relevant equations
Conservation of Energy , s = R*a, Newton's Second Law
3. The attempt at a solution
a) Forces are static friction, normal force and gravity
b) Applying conservation of energy gives v^2 = 4/3 ((R+r)(1cos(a))g. Since the wheel falls off when N =0, a = arccos(4/7). Therefore s = R*arccos(4/7).
c) Conservation of energy gives 0.5m (v_f^2  v_c^2) = mg((4/7 R+3R)). Therefore v_f^2 = 2g(25/7 R) +4/7 (R+r)g.
