1. AP Physics B Summer Course 2012 2012 年 AP 物理 B 暑假班 M Sittig Ch 15: Gravitation and Circular Motion

2. Uniform Circular Motion

3. Centripetal Acceleration  Centripetal = “toward the center”

4. Centripetal acceleration Centripetal acceleration (m/s 2 ) Radius (m) Tangential velocity (m/s)

5. Centripetal force  Centripetal force is whatever force is causing the centripetal acceleration.  This can be the net force, or some other force: friction, gravity, tension, magnetic force…  And a useful shortcut:

6. Three examples  Mass on a string  Car on a curve  Friction  Banked  Satellite in orbit

7. Mass on a String

8. Practice Problem  A girl whirls a stone in a horizontal circle 2.1 m above the ground by means of a string 154 cm long. The string breaks, and the stone flies of horizontally and strikes the ground 11.3 m away. What was the centripetal acceleration of the stone while it was in circular motion?

9. Practice Problem  To study circular motion, two students use the hand ‑ held device shown above, which consists of a rod on which a spring scale is attached. A polished glass tube attached at the top serves as a guide for a light cord attached the spring scale. A ball of mass 0.200 kg is attached to the other end of the cord. One student swings the ball around at constant speed in a horizontal circle with a radius of 0.500 m. Assume friction and air resistance are negligible.

10. Practice Problem  a. Explain how the students, by using a timer and the information given above, can determine the speed of the ball as it is revolving.  b. How much work is done by the cord in one revolution? Explain how you arrived at your answer.

11. Practice Problem  c. The speed of the ball is determined to be 3.7 m/s. Assuming that the cord is horizontal as it swings, calculate the expected tension in the cord.  d. The actual tension in the cord as measured by the spring scale is 5.8 N. What is the percent difference between this measured value of the tension and the value calculated in part c. ?

12. Practice Problem  e. The students find that, despite their best efforts, they cannot swing the ball so that the cord remains exactly horizontal.  i. Draw a free body diagram showing the forces acting on the ball and identify the force that each vector represents. ii. Explain why it is not possible for the ball to swing so that the cord remains exactly horizontal.  iii. Calculate the angle that the cord makes with the horizontal.

13. Car on a Curve

14. Practice Problem  A turn of radius 100 m is being designed for a speed of 25 m/s. At what angle should the turn be banked so that a car can make the turn even under conditions of no friction?

15. Satellite in Orbit  First, examine Newton’s Law of Gravitation:

16. Newton’s Law of Gravitation Force of gravity (m/s 2 ) Radius (m) Universal gravitational constant (m 3 /kg ·s 2 ) Mass (kg)

17. Satellite in Orbit  But I thought the force of gravity on an object (weight) is F = mg???  Try calculating G·M 1 /r 2 with the mass and radius of the Earth…  So F = (G M 1 M 2 )/r 2 = M 2 ·g

18. Example Problem

19. Practice Problem  At what distance above the surface of the Earth should a geostationary satellite orbit?

20. Practice Problem  A planet is a uniform sphere of mass M. The length of the day on the planet (that is, the period of rotation about its axis) is T. The linear speed of the points on the equator of the planet is v. Based on this information, what would be the reading (in newtons) of the bathroom scale on the equator if a person of mass m stands on it?

21. Gravitational Potential Energy  Not used often, but know it.

22. Gravitational Potential Energy Potential Energy (J) Radius (m) Universal gravitational constant (m 3 /kg ·s 2 ) Mass (kg)

23. Gravitational Potential Energy  Wait, isn’t PE g = mgh? Two answers:  No, that’s difference in PE.  Yes, remember that g = G·M E /R E 2, so if we take the center of the Earth to be PE = 0, then PE g = PE G. Also, this new equation works for any two bodies.