The kinematics of uniform circular motion The dynamics of uniform circular motion

1. Chapter 6 Circular Motion, Orbits and Gravity • The kinematics of uniform circular motion • The dynamics of uniform circular motion • Circular orbits of satellites • Newton’s law of gravity Topics: Sample question: The motorcyclist in the “Globe of Death” rides in a vertical loop upside down over the top of a spherical cage. There is a minimum speed at which he can ride this loop. How slow can he go? Slide 6-1

2. Kick the Ball • If the UNM kicker kicks the ball with an initial velocity of 25 m/s at an angle of 45 degrees with the horizontal, • How far does the ball land down the field? • How high does the ball go? • How long is the ball in the air? • Bonus - in symbols, how does the time for the ball to go up compare to the time it takes to come down? Slide 6-14

3. Going in Circles An object makes 10 revolutions going around a circle in 2 seconds. How long does each revolution take? What is the frequency of revolutions? If the circle has a radius r = 30 cm, what is the object's speed? Slide 6-14

4. Uniform Circular Motion Slide 6-13

5. v = r v2 r A = — = 2 r Fnet = ma = {—, toward center of circle} mv2 r   Forces in Circular Motion Slide 6-21

6. Force Analysis Examples • Ball on String on table • Ball with plastic circle • Ball on string hanging • Ball on string in vertical circle Slide 6-14

7. Circular Motion Dynamics When the ball reaches the break in the circle, which path will it follow? Slide 6-19

8. Answer When the ball reaches the break in the circle, which path will it follow? Slide 6-20

9. Solving Problems Slide 6-22

10. Example In the hammer throw, an athlete spins a heavy mass in a circle at the end of a chain, then lets go of the chain. For male athletes, the “hammer” is a mass of 7.3 kg at the end of a 1.2 m chain. A world-class thrower can get the hammer up to a speed of 29 m/s. If an athlete swings the mass in a horizontal circle centered on the handle he uses to hold the chain, what is the tension in the chain? Slide 6-25

11. Driving over a Rise • A car of mass 1500 kg goes over a hill at a speed of 20 m/s. The shape of the hill is approximately circular, with a radius of 60 m, as in the figure at right. When the car is at the highest point of the hill, • What is the force of gravity on the car? • What is the normal force of the road on the car at this point? • What is the maximum speed the care can have as it goes over the hill without leaving the ground? Slide 6-26

12. Bug on a CD A bug crawls outward from the center of a compact disc spinning at 175
revolutions per minute. The coefficient of static friction between the bug's sticky
feet and disk is 0.9.How far does the bug get from the center before slipping off ? Slide 6-14

13. Translational Motion vs. Rotational Motion Slide 6-14

14. Reading Quiz • For uniform circular motion, the acceleration • is parallel to the velocity. • is directed toward the center of the circle. • is larger for a larger orbit at the same speed. • is always due to gravity. • is always negative. Slide 6-2

15. Answer • For uniform circular motion, the acceleration • is directed toward the center of the circle. Slide 6-3

16. Reading Quiz • When a car turns a corner on a level road, which force provides the necessary centripetal acceleration? • Friction • Tension • Normal force • Air resistance • Gravity Slide 6-4

17. Answer • When a car turns a corner on a level road, which force provides the necessary centripetal acceleration? • Friction Slide 6-5

18. Checking Understanding When a ball on the end of a string is swung in a vertical circle: • We know that the ball is accelerating because • the speed is changing. • the direction is changing. • the speed and the direction are changing. Slide 6-9

19. Answer When a ball on the end of a string is swung in a vertical circle: • We know that the ball is accelerating because • the direction is changing. Slide 6-10

20. Checking Understanding When a ball on the end of a string is swung in a vertical circle: • What is the direction of the acceleration of the ball? • Tangent to the circle, in the direction of the ball’s motion • Toward the center of the circle Slide 6-11

21. Answer When a ball on the end of a string is swung in a vertical circle: • What is the direction of the acceleration of the ball? • Toward the center of the circle Slide 6-12

22. Examples The disk in a hard drive in a desktop computer rotates at 7200 rpm. The disk has a diameter of 5.1 in (13 cm.) What is the angular speed of the disk? The hard drive disk in the previous example rotates at 7200 rpm. The disk has a diameter of 5.1 in (13 cm.) What is the speed of a point 6.0 cm from the center axle? What is the acceleration of this point on the disk? Slide 6-14

23. Circular Motion Dynamics For the ball on the end of a string moving in a vertical circle: • What force is producing the centripetal acceleration of the ball? • gravity • air resistance • normal force • tension in the string Slide 6-15

24. Answer For the ball on the end of a string moving in a vertical circle: • What force is producing the centripetal acceleration of the ball? • tension in the string Slide 6-16

25. Circular Motion Dynamics For the ball on the end of a string moving in a vertical circle: • What is the direction of the net force on the ball? • tangent to the circle • toward the center of the circle • there is no net force Slide 6-17

26. Answer For the ball on the end of a string moving in a vertical circle: • What is the direction of the net force on the ball? • toward the center of the circle Slide 6-18

27. Slide 6-23

28. Example A level curve on a country road has a radius of 150 m. What is the maximum speed at which this curve can be safely negotiated on a rainy day when the coefficient of friction between the tires on a car and the road is 0.40? Slide 6-24

29. Maximum Walking Speed Slide 6-27

30. Loop-the-Loop A roller coaster car goes through a vertical loop at a constant speed. For positions A to E, rank order the: • centripetal acceleration • normal force • apparent weight Slide 6-28

31. Over the Top A handful of professional skaters have taken a skateboard through an inverted loop in a full pipe. For a typical pipe with diameter 14 ft, what is the minimum speed the skater must have at the very top of the loop? Slide 6-29