In problems dealing with a pulley with a nonzero moment of

Chapter , Problem 127

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In problems dealing with a pulley with a nonzero moment of inertia, the magnitude of the tensions in the ropes hanging on either side of the pulley are not equal. The difference in the tension is due to the static frictional force between the rope and the pulley; however, the static frictional force cannot be made arbitrarily large. Consider a massless rope wrapped partly around a cylinder through an angle \(\Delta \theta\) (measured in radians). It can be shown that if the tension on one side of the pulley is T, while the tension on the other side is \(T^{\prime}\left(T^{\prime}>T\right)\) the maximum value of \(T^{\prime}\) that can be maintained without the rope slipping is \(T_{\max }^{\prime}=T e^{\mu_{s} \Delta \theta}\) where \(\mu_{\mathrm{s}}\) is the coefficient of static friction. Consider the Atwood’s machine in Figure 9-76: the pulley has a radius the moment of inertia is \(I=0.35 \mathrm{\ kg} \cdot \mathrm{m}^{2}\) and the coefficient of static friction between the wheel and the string is \(\mu_{\mathrm{s}}=0.30\) (a) If the tension on one side of the pulley is what is the maximum tension on the other side that will prevent the rope from slipping on the pulley? (b) What is the acceleration of the blocks in this case? (c) If the mass of one of the hanging blocks is what is the maximum mass of the other block if, after the blocks are released, the pulley is to rotate without slipping?