14. Torque & Rotational Dynamics

A hollow cylinder (hoop) is rolling on a horizontal surface at speed v = 3.0 m/s when it reaches an 18° incline.

(a) How far up the incline will it go?

A 1.6-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires a rotational rate of 22 rev/s from rest over a 6.0-s interval at constant angular acceleration. Calculate the torque delivered by the motor.

Suppose David puts a 0.60-kg rock into a sling of length 1.5 m and begins whirling the rock in a nearly horizontal circle, accelerating it from rest to a rate of 75 rpm after 4.5 s. What is the torque required to achieve this feat, and where does the torque come from?

(II) The forearm in Fig. 10–57 accelerates a 3.6-kg ball at 7.0 m/s² by means of the triceps muscle, as shown. Calculate

(a) the torque needed,

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(II) The forearm in Fig. 10–57 accelerates a 3.6-kg ball at 7.0 m/s² by means of the triceps muscle, as shown. Calculate

(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm

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(II) A solid rubber ball rests on the floor of a railroad car when the car begins moving with acceleration a. Assuming the ball rolls without slipping, what is its acceleration relative to

(a) the car and ?

(II) A rotating uniform cylindrical platform of mass 220 kg and radius 5.5 m slows down from 3.8 rev/s to rest in 18 s when the driving motor is disconnected. Estimate the power output of the motor (hp) required to maintain a steady speed of 3.8 rev/s .

Water drives a waterwheel (or turbine) of radius R = 3.0 m as shown in Fig. 11–50. The water enters at a speed v₁ = 7.0m/s and exits from the waterwheel at a speed v₂= 3.8 m/s.

(c) If the water causes the waterwheel to make one revolution every 6.0 s, how much power is delivered to the wheel?

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If a billiard ball is hit in just the right way by a cue stick, the ball will roll without slipping immediately after losing contact with the stick. Consider a billiard ball (radius r, mass M) at rest on a horizontal pool table. A cue stick exerts a constant horizontal force F on the ball for a time t at a point that is a height h above the table’s surface (see Fig. 10–78). Assume that the coefficient of static friction between the ball and table is μₛ . Determine the value for h so that the ball will roll without slipping immediately after losing contact with the stick.

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"A boy rolls a tire along a straight level street. The tire has mass 8.0 kg, radius 0.32 m and moment of inertia about its central axis of symmetry of 0.83 kg·m². The boy pushes the tire forward away from him at a speed of 2.1 m/s and sees that the tire leans 12° to the right (Fig. 11–49).                                                                                                                           

 (a) How will the resultant torque due to gravity and the normal force F→_N affect the subsequent motion of the tire? 

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(II) Assume that a 1.00-kg ball is thrown solely by the action of the forearm, which rotates about the elbow joint under the action of the triceps muscle, Fig. 10–57. The ball is accelerated uniformly from rest to 8.5 m/s in 0.38 s, at which point it is released. Calculate

(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.7 kg and rotates like a uniform rod about an axis at its end.

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A particle of mass m uniformly accelerates as it moves counterclockwise along the circumference of a circle of radius R:

r→ = î R cos θ + ĵ R sin θ

with θ = ω₀t + (1/2)αt² , where the constants ω₀ and α are the initial angular velocity and angular acceleration, respectively. Determine the object’s tangential acceleration a→_tan and determine the torque acting on the object using 

(b) τ→ = Iα→ .

(II) To get a flat, uniform cylindrical satellite spinning at the correct rate, engineers fire four tangential rockets as shown in Fig. 10–61. Suppose that the satellite has a mass of 3600 kg and a radius of 4.0 m, and that the rockets each add a mass of 250 kg. What is the steady force required of each rocket if the satellite is to reach 28 rpm in 5.0 min, starting from rest?

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(II) A solid rubber ball rests on the floor of a railroad car when the car begins moving with acceleration a. Assuming the ball rolls without slipping, what is its acceleration relative to

(b) the ground?

(II) Calculate the moment of inertia of the array of point objects shown in Fig. 10–58 about the y axis, and the x axis. Assume m = 22kg, M = 3.2kg, and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the x axis.

(c) About which axis would it be harder to accelerate this array?

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(II) A dad pushes tangentially on a small hand-driven merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 330 kg, and two children (each with a mass of 25 kg) sit opposite each other on the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. What force is required at the edge?