14. Torque & Rotational Dynamics

The flywheel of an engine has moment of inertia 1.60 kg/m² about its rotation axis. What constant torque is required to bring it up to an angular speed of 400 rev/min in 8.00 s, starting from rest?

A softball player swings a bat, accelerating it from rest to 2.4 rev/s in a time of 0.20 s. Approximate the bat as a 0.90-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.

A large spool of rope rolls on the ground with the end of the rope lying on the top edge of the spool. A person grabs the end of the rope and walks a distance ℓ, holding onto it, Fig. 10–70. The spool rolls behind the person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

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A 12-cm-diameter, 600 g cylinder, initially at rest, rotates on an axle along its axis. A steady 0.50 N force applied tangent to the edge of the cylinder causes the cylinder to reach an angular velocity of 500 rpm in 2.0 s. What is the magnitude of the frictional torque between the cylinder and the axle?

Your engineering team has been assigned the task of measuring the properties of a new jet-engine turbine. You've previously determined that the turbine's moment of inertia is 2.6 kg m². The next job is to measure the frictional torque of the bearings. Your plan is to run the turbine up to a predetermined rotation speed, cut the power, and time how long it takes the turbine to reduce its rotation speed by 50%. Your data are given in the table. Draw an appropriate graph of the data and, from the slope of the best-fit line, determine the frictional torque.

A 30-cm-diameter, 1.2 kg solid turntable rotates on a 1.2-cm-diameter, 450 g shaft at a constant 33 rpm. When you hit the stop switch, a brake pad presses against the shaft and brings the turntable to a halt in 15 seconds. How much friction force does the brake pad apply to the shaft?

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

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 the torque needed.

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

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 the car?

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

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 μ_s. Determine the value for h so that the ball will roll without slipping immediately after losing contact with the stick.