AP Physics 1 Chapter 7 Circular Motion and Gravitation

1. AP Physics 1 Chapter 7Circular Motion and Gravitation

2. Chapter 7: Circular Motion and Gravitation Angular Measure Angular Speed and Velocity Uniform Circular Motion and Centripetal Acceleration Angular Acceleration Newton’s Law of Gravitation Kepler’s Laws and Earth Satellites

3. Homework for Chapter 7 Read Chapter 7 • Chapter 7:  #s 15,16,17,21,23,25,26,27,29,51  • #18,19,22,30,31,32,54,55,60,65,66,68,70, • # 62,63,66,68,70, 71,75 • # 33,35,36,37,38,40 • #34,39,56,57,59#41-47,53

4. Angular MeasureAngular Speed and Velocity

5. Angular Measure The relationship between rectangular coordinates and polar coordinates are: x = r cos Ө y = r sin Ө rotation – axis of rotation lies within the body (example: Earth rotates on its axis) revolution – axis of rotation lies outside the body (example: Earth revolves around the Sun) • Circular motion is conveniently described using polar coordinates (r,Ө) because r is a constant and only Ө varies. • Ө is measured counter-clockwise from the +x axis.

6. Angular Measure Angular distance (∆Ө = Ө – Ө0) may be measured in either degrees or radians (rad). 1 rad ≈ 57.3° or 2𝜋 rad = 360°

7. Angular Measure

8. Angular Measure

9. Angular Measure Example 7.1: When you are watching the NASCAR Daytona 500, the 5.5 m long race car subtends and angle of 0.31°. What is the distance from the race car to you?

10. Angular Speed and Velocity Linear analogy: a = ∆ v ∆ t Linear analogy: v = ∆ x ∆ t

11. Angular Speed and Velocity The units of angular acceleration are rad/s2. The way to remember this is the right-hand rule: When the fingers of the right hand are curled in the direction of rotation, the extended thumb points in the direction of the angular velocity or angular acceleration vector.

12. Angular Speed and Velocity • Tangential and angular speeds are related by v = rω, with ωin radians per second. Note, all of the particles rotating about a fixed axis travel in circles. All of the particles have the same angular speed (ω). Particles at different distances from the axis of rotation have different tangential speeds. • Sparks from a grinding wheel illustrate instantaneous tangential velocity.

13. Angular Speed and Velocity For every linear quantity or equation there is an analogous angular quantity or equation. (Assume x0 = 0, θ0 = 0, t0 = 0). Substitute θ→ x, ω→ v, α→ a.

14. Angular Speed and Velocity • When angular speed and velocity are given in units of rpm (revolutions per minute) you should first convert them to rad/s before trying to solve the problem. Example 7.2a: Convert 33 rpm to rad/s.

15. Angular Speed and Velocity f = frequency T = period ω = angular speed

16. Angular Speed and Velocity • The SI unit of frequency is 1/sec or hertz (Hz).

17. Angular Speed and Velocity • Example 7.2b: A bicycle wheel rotates uniformly through 2.0 revolutions in 4.0 s. • What is the average angular speed of the wheel? • What is the tangential speed of a point 0.10 m from the center of the wheel? • What is the period? • What is the frequency?

18. Check for Understanding

19. Check for Understanding

20. Check for Understanding

21. Check for Understanding

22. Uniform Circular Motion andCentripetal AccelerationAngular Acceleration

23. Uniform Circular Motion and Centripetal Acceleration Physics Warmup # 35

24. Uniform Circular Motion and Centripetal Acceleration Physics Warmup # 35

25. Uniform Circular Motion and Centripetal Acceleration uniform circular motion An object moves at a constant speed in a circular path. The speed of an object in uniform circular motion is constant, but the object’s velocity changes in the direction of motion. Therefore, there is an acceleration. Fig. 7.8 p.218

26. Uniform Circular Motion and Centripetal Acceleration centripetal acceleration – center-seeking For and object in uniform circular motion, the centripetal acceleration is directed towards the center. There is no acceleration component in the tangential direction. If there were, the magnitude of the velocity (tangential speed) would change. ac = v2 = rω2 r Fig. 7.10, p.219

27. Uniform Circular Motion and Centripetal Acceleration • From Newton’s second law, Fnet = ma. Therefore, there must be a net force associated with centripetal acceleration. • In the case of uniform circular motion, this force is called centripetal force. It is always directed toward the center of the circle since we know the net force on an object is in the same direction as acceleration. Fc = mac = mv2= mrω2 r • Centripetal force is not a separate or extra force. It is a net force toward the center of the circle. • A centripetal force is always required for objects to stay in a circular path. Without it, an object will fly out along a tangent line due to inertia.

28. Uniform Circular Motion and Centripetal Acceleration • The time period T, the frequency of rotation f, the radius of the circular path, and the speed of the particle undergoing uniform circular motion are related by: T = 2 π r = 1 = 2 π v f ω centrifugal force – center-fleeing force; a fictitious force; something made up by nonphysicists; the vector equivalent of a unicorn Hint: Do not label a force as “centripetal force” on your free-body diagram even if that force does act toward the center of the circle. Rather, label the actual source of the force; i.e., tension, friction, weight, electric force, etc. Question 1: What provides the centripetal force when clothes move around a dryer? (the inside of the dryer) Question 2: What provides the centripetal force upon a satellite orbiting the Earth? (Earth’s gravity)

29. Uniform Circular Motion and Centripetal Acceleration Example 7.a:

30. Uniform Circular Motion and Centripetal Acceleration • Example 7.3: A car of mass 1500 kg is negotiating a flat circular curve of radius 50 meters with a speed of 20 m/s. • What is the source of centripetal force on the car? • What is the magnitude of the centripetal acceleration of the car? • What is the magnitude of the centripetal force on the car?

31. Uniform Circular Motion and Centripetal Acceleration Example 7.3a: A car approaches a level, circular curve with a radius of 45.0 m. If the concrete pavement is dry, what is the maximum speed at which the car can negotiate the curve at a constant speed?

32. Uniform Circular Motion and Centripetal Acceleration Check for Understanding: 1. In uniform circular motion, there is a a. constant velocity b. constant angular velocity c. zero acceleration d. net tangential acceleration Answer: b

33. Uniform Circular Motion and Centripetal Acceleration Check for Understanding: 2. If the centripetal force on a particle in uniform circular motion is increased, a. the tangential speed will remain constant b. the tangential speed will decrease c. the radius of the circular path will increase d. the tangential speed will increase and/or the radius will decrease Answer: d; Fc = mv2 r

34. Uniform Circular Motion and Centripetal Acceleration Check for Understanding: 3. Explain why mud flies off a fast-spinning wheel. Answer: Centripetal force is proportional to the square of the speed. When there is insufficient centripetal force (provided by friction and adhesive forces), the mud cannot maintain the circular path and it flies off along a tangent.

35. Angular Acceleration • Average angular acceleration () is  =    t • The SI unit of angular acceleration is rad/s2. • The relationship between tangential and angular acceleration is at = r  (This is not to be confused with centripetal acceleration, ac).

36. Angular Acceleration

37. Angular Acceleration

38. In uniform circular motion, there is centripetal acceleration but no angular acceleration (α = 0) or tangential acceleration (at = r α = 0). In nonuniform circular motion, there are angular and tangential accelerations. at = ∆v = ∆(rω) = r∆ω = rα ∆t ∆t ∆t Fig. 7.16, p.226

39. 7.4 Angular Acceleration • There is always centripetal acceleration no matter whether the circular motion is uniform or nonuniform. • It is the tangential acceleration that is zero in uniform circular motion. • Example 7.4: A wheel is rotating wit a constant angular acceleration of 3.5 rad/s2. If the initial angular velocity is 2.0 rad/s and is speeding up, find • the angle the wheel rotates through in 2.0 s • the angular speed at t = 2.0 s

40. Angular Acceleration Example 7.5: The power on a medical centrifuge rotating at 12,000 rpm is cut off. If the magnitude of the maximum deceleration of the centrifuge is 50 rad/s2, how many revolutions does it rotate before coming to rest?

41. Angular Acceleration Check for Understanding: 1. The angular acceleration in circular motion a. is equal in magnitude to the tangential acceleration divided by the radius b. increases the angular velocity if in the same direction c. has units of rad/s2 d. all of the above Answer: d

42. Angular Acceleration Check for Understanding: 2. Can you think of an example of a car having both centripetal acceleration and angular acceleration? Answer: Yes, when a car is changing its speed on a curve.

43. Angular Acceleration Check for Understanding: 3. Is it possible for a car in circular motion to have angular acceleration but not centripetal acceleration? Answer: No, this is not possible. Any car in circular motion always has centripetal acceleration.

44. Newton’s Law of GravitationKepler’s Laws and Earth Satellites

45. Uniform Circular Motion and Centripetal Acceleration Physics Warmup # 48 Solution: It would decrease. You would have mass below you pulling downward and mass above you pulling upward. At the center of the earth, you would weigh zero.

46. Newton’s Law of Gravitation G is the universal gravitational constant.

47. Newton’s Law of Gravitation F α1 Inverse Square Law r2

48. For homogeneous spheres, the masses may be considered to be concentrated at their centers. Any two particles, or point masses, are gravitationally attracted to each other with a force that has a magnitude given by Newton’s universal law of gravitation. Fig. 7.17, p.228

49. Newton’s Law of Gravitation

50. Newton’s Law of Gravitation