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Weight and Wheel
Consider a bicycle wheel that initially is not rotating. A block of mass m is attached to the wheel and is allowed to fall a distance h. Assume that the wheel has a moment of inertia I about its rotation axis.
Part A  
Consider the case that the string tied to the block is attached to the outside of the wheel, at a radius r_A

. Find omega_A, the angular speed of the wheel after the block has fallen a distance h, for this case.
Hint A.1  

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Part A.2  

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Part A.3  

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Part A.4  

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Express omega_A in terms of m, g, h, r_A, and I.
ANSWER:
  omega_A =  (2*m*g*h/(m*r_A^2+I))^(1/2) 
Part B  
Now consider the case that the string tied to the block is wrapped around a smaller inside axle of the wheel of radius r_B . Find omega_B, the angular speed of the wheel after the block has fallen a distance h, for this case.
Hint B.1  

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Express omega_B in terms of m, g, h, r_B, and I.
ANSWER:
  omega_B =  (2*m*g*h/(m*r_B^2+I))^(1/2) 
Part C  
Which of the following describes the relationship between omega_A and omega_B?
Hint C.1  

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ANSWER:
\omega_A > \omega_B
\omega_B > \omega_A
\omega_A = \omega_B
This is related to why gears are found on the inside rather than the outside of a wheel.
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