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Grade 11Mechanics

Two wheels, A and B, are connected by a belt as in Fig. 11-29. The radius of B is three times the radius of A. What would be the ratio of the rotational inertias IA/IB, if (a) both wheels have the same angular momenta and (b) both wheels have the same rotational kinetic energy? Assume that the belt does not slip.
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

Profile image of Radhika Batra
11 Years agoGrade 11
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1 Answer

Profile image of Kevin Nash
11 Years ago
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From the above observation we conclude that, if both wheels have the same rotational kinetic energy, then the ratio of rotational inertia would be \frac{1}{9}.