Problem 10.76 A thin-walled, hollow spherical shell of mass m and radius r start

Question

Problem 10.76 A thin-walled, hollow spherical shell of mass m and radius r starts from rest and rolls without slipping down the track shown in the figure (Figure 1) . Points A and B are on a circular part of the track having radius R. The diameter of the shell is very small compared to h0 and R, and the work done by the rolling friction is negligible.

Part A What is the minimum height h0 for which this shell will make a complete loop-the-loop on the circular part of the track? Express your answer in terms of the variables m, R, and appropriate constants.

Part B How hard does the track push on the shell at point B, which is at the same level as the center of the circle? Express your answer in terms of the variables m, R, and appropriate constants.

Part C Suppose that the track had no friction and the shell was released from the same height h0 you found in part (a). Would it make a complete loop-the-loop?

Part D In part (c), how hard does the track push on the shell at point A, the top of the circle? Express your answer in terms of the variables m, R, and appropriate constants.

Part E How hard did the track push on the shell at point A in part (a)? Express your answer in terms of the variables m, R, and appropriate constants.

Explanation / Answer

I=2*m*r^2/3

at the top of the loop
v^2/R=g
v^2=R*g

m*g*(h-2*R)=
.5*m*v^2+.5*2*m*r^2*v^2/(3*r^2)

simplify

2*g*(h-2*R)=5*v^2/3
plug in R*g

2*g*(h-2*R)=5*R*g/3

solve for h

h=R*13/6

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