# Why does a styrofoam coffee cup roll in a circle?

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1 Why does a styrofoam coffee cup roll in a circle? Why does a styrofoam coffee cup roll in a circle? 1

2 2

3 Example 1. The blades of an electric blender are whirling with an angular velocity of +375 rad/s while the puree button is pushed in. When the blend button is pressed, the blades accelerate and reach a greater angular velocity after the blades have rotated through an angular displacement of rad (seven revolutions). The angular acceleration has a constant value of rad/s 2. Find the final angular velocity of the blades. 3

4 Example 2. The blades of a ceiling fan start from rest and, after two revolutions, have an angular speed of 0.50 rev/s. The angular acceleration of the blades is constant. What is the angular speed after eight revolutions? 4

5 Example 3. A centrifuge rotor is accelerated from rest to 200,000 rpm in 30 s. a. What is the average angular acceleration? b. What is the period of revolution? 5

6 6

7 Example 4. An automobile starts from rest and for 20.0 s has a constant linear acceleration of m/s 2 to the right. During this period, the tires do not slip. The radius of the tires is m. At the end of the 20.0 s interval, what is the angle through which each wheel has rotated? 7

8 Example 5. A rider on a mountain bike is traveling to the left. Each wheel has an angular velocity of rad/s, where, as usual, the plus sign indicates that the wheel is rotating in the counterclockwise direction. In both instances below, determine the magnitude and direction of the angular acceleration (assumed constant) of the wheels. a) To pass another cyclist, the rider pumps harder, and the angular velocity of the wheels increases from to rad/s in a time of 3.50 s. b) After passing the cyclist, the rider begins to coast, and the angular velocity of the wheels decreases from to rad/s in a time of 10.7 s. 8

9 Example 6. Suppose you are driving a car in a counterclockwise direction on a circular road whose radius is r = 390 m. You look at the speedometer and it reads a steady 32 m/s (about 72 mi/h). (a) What is the angular speed of the car? (b) To avoid a rear end collision with a vehicle ahead, you apply the brakes and reduce your angular speed to rad/s in a time of 4.0 s. What is the tangential acceleration (magnitude and direction) of the car? 9

10 Example 7. A bicycle slows down uniformlly from v o = 8.4 m/s to rest over a distance of 115 m. Each wheel and tire has an overall diameter of 68.0 cm. Determine a) the initial angular velocity of the wheels. b) the total number of revolutions the wheels make before coming to rest. c) the angular acceleration of the wheel. d) the time it takes to come to a complete stop. This is Example 8 7 on page 202 of the book. 10

11 P , 17, (I) A centrifuge accelerates uniformly from rest to 15,000 rpm in 220 s. Through how many revolutions did it turn in this time? 16. (I) An automobile engine slows down from 4500 rpm to 1200 rpm in 2.5 s. Calculate (a) its angular acceleration. (b) the total number of revolutions the engine makes in this time. 17. (I) Pilots can be tested for the stresses of flying highspeed jets in a whirling human centrifuge, which takes 1.0 min to turn through 20 complete revolutions before reaching its final speed. (a) What was its angular acceleration? (b) What was its final angular speed in rpm? 11

12 18. (II) A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 s. How far will a point on the edge of the wheel have traveled in this time? 19. (II) A cooling fan is turned off when it is running at 850 rev/min. It turns 1500 revolutions before it comes to a stop. (a) What was the fan s angular acceleration, assumed constant? (b) How long did it take the fan to come to a complete stop? 20. (II) A small rubber wheel is used to drive a large pottery wheel, and they are mounted so that their circular edges touch. The small wheel has a radius of 2.0 cm and accelerates at the rate of 7,2 rad/s 2, and it is in contact with the pottery wheel (radius 25.0 cm) without slipping. Calculate (a) the angular acceleration of the pottery wheel. (b) the time it takes the pottery wheel to reach its required speed of 65 rpm. 12

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