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Figure P10.63 Problems 63 and 64.

64.A bicycle is turned upside down while its owner repairs a flat tire. A friend spins the other wheel of radius R and observes that drops of water fly off tangentially. She measures the height reached by drops moving vertically (Fig.  P10.63). A drop that breaks loose from the tire on one turn rises a distance h1 above the tangent point. A drop that breaks loose on the next turn rises a distance h2 < h1 above the tangent point. The height to which the drops rise decreases because the angular speed of the wheel decreases. From this information, determine the magnitude of the average angular acceleration of the wheel.

65.A cord is wrapped around a pulley of mass m and radius r. The free end of the cord is connected to a block of mass M. The block starts from rest and then slides down an incline that makes an angle with the horizontal. The coefficient of kinetic friction between block and incline is. (a) Use energy methods to show that the block's speed as a function of position d down the incline is

(b) Find the magnitude of the acceleration of the block in terms of , m, M, g, and .

66. (a) What is the rotational kinetic energy of the Earth about its spin axis? Model the Earth as a uniform sphere and use data from the endpapers. (b) The rotational kinetic energy of the Earth is decreasing steadily because of tidal friction. Find the change in one day, assuming that the rotational period decreases by 10.0 s each year.

67. Due to a gravitational torque exerted by the Moon on the Earth, our planet’s rotation period slows at a rate on the order of 1 ms/century. (a) Determine the order of magnitude of the Earth’s angular acceleration. (b) Find the order of magnitude of the torque. (c) Find the order of magnitude of the size of the wrench an ordinary person would need to exert such a torque, as in Figure P10.67.  Assume the person can brace his feet against a solid firmament.

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