In a charming 19th-century hotel, an old-style elevator is connected to a counterweight by a cable that passes over a rotating disk 3.50 m in diameter. The elevator is raised and lowered by turning the disk, and the cable does not slip on the rim of the disk but turns with it. (a) At how many rpm must the disk turn to raise the elevator at 25.0 cm/s? (b) To start the elevator moving, it must be accelerated at 19g. What must be the angular acceleration of the disk, in rad/s2?

Astronauts in the International Space Station must work out every day to counteract the effects of weightlessness. Researchers have investigated if riding a stationary bicycle while experiencing artificial gravity from a rotating platform gives any additional cardiovascular benefit. What frequency of rotation, in rpm, is required to give an acceleration of 1.4g to an astronaut’s feet, if her feet are 1.1 m from the platform’s rotational axis?

A 1.3 kg rock is swung in a circular path and in a vertical plane on a 1 m length string. At the top of the path, the angular velocity is 9 rad/s. What is the tension in the string at that point?

The royal antelope of western Africa has an average mass of only 2.5 kg. Suppose thisantelope runs in a circle with a radius of 35.0 m. If a force of 8.8 N maintains this circularmotion, what is the antelope’s tangential speed?

The photograph below shows a girl pushing on a large stone sphere. The sphere has a mass of 8200 kg and a radius of 90 cm and floats with nearly zero friction on a thin layer of pressurized water. Suppose that she pushes on the sphere tangent to its surface with a steady force of F = 25 N and that the pressured water provides a frictionless support. How long will it take her to rotate the sphere one time, starting from rest?

You have a horizontal grindstone (a disk) that is 87 kg, has a 0.31 m radius, is turning at 94 rpm (in the positive direction), and you press a steel axe against the edge with a force of 23 N in the radial direction. The kinetic coefficient of friction between steel and stone is 0.20. What is the number of turns, N, that the stone will make before coming to rest?

A van with mass m = 1000 kg is driving around a circular curve of radius r = 950 m when it begins to rain. The coefficients of static friction between the track and tires is μd = 0.59 when dry and μw = 0.17 when wet. Write an expression for the maximum magnitude of the force of static friction Ff acting on the van if μs is the coefficient of friction. What is the maximum tangential speed that the van can take the corner at without slipping when dry in m/s? What is the maximum tangential speed when wet in m/s?

The motion of spinning a hula hoop around one's hips can be modeled as a hoop rotating around an axis not through the center, but offset from the center by an amount h, where h is less than R, the radius of the hoop. Suppose Maria spins a hula hoop with a mass of 0.74 kg and a radius of 0.67 m around her waist. The rotation axis is perpendicular to the plane of the hoop, but approximately 0.45 m from the center of the hoop. (a) What is the rotational inertia of the hoop in this case? 0.45 X Your response is within 10% of the correct value. This may be due to roundoff error, or you could have a mistake in your calculation. Carry out all intermediate results to at least four-digit accuracy to minimize roundoff error. kg. m² (b) If the hula hoop is rotating with an angular speed of 13.3 rad/s, what is its rotational kinetic energy? 0.74 X Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. J

You have a disk that is 105 kg, has a 0.297-m radius, and is turning at 71.15 rpm, and you press a steel axe against it with a radial force of 46.65 N. Assuming the kinetic coefficient of friction between steel and stone is 0.451. How many turns will the stone make before coming to rest?

A coin rests 15.0 cm from the center of a turn-table. The coefficient of static friction between the coin and turntable surface is 0.350.  The turntable starts from rest at t = 0 and rotates with a constant angular acceleration of 0.730 rad/s2.  (A) Once the turntable starts to rotate, what force causes the centripetal acceleration when the coin is stationary relative to the turntable? Under what condition does the coin begin to move relative to the turntable? (B) After what period of time will the coin start to slip on the turntable?