Unit 3: Motion and Forces in 2 and 3 Dimensions

1. Unit 3: Motion and Forces in 2 and 3 Dimensions • A) Motion in Circles • B) Centripetal Force • C) Universal Gravitation and Orbital Motion

2. Key Points • Calculate angular speed in radians per second. • Calculate linear speed from angular speed and vice-versa. • Describe and calculate centripetal forces and accelerations. • Describe the relationship between the force of gravity and the masses and distance between objects. • Calculate the force of gravity when given masses and distance between two objects. • Describe why satellites remain in orbit around a planet.

3. rotate • revolve • axis • law of universal gravitation • circumference • linear speed • angular speed • centrifugal force • radian • orbit • centripetal force • centripetal acceleration • ellipse • satellite • angular displacement • gravitational constant Vocabulary

4. Key Question: How do we describe circular motion? Vectors and Direction

5. Motion in Circles • We say an object rotates about its axis when the axis is part of the moving object. • A child revolves on a merry-go-round because he is external to the merry-go-round's axis.

6. Angular Speed • Angular speed is the rate at which an object rotates or revolves. • There are two ways to measure angular speed • number of turns per unit of time (rotations/minute) • change in angle per unit of time (deg/sec or rad/sec)

7. Angular Speed • For the purpose of angular speed, the radian is a better unit for angles. • One radian is approx. 57.3 degrees. • Radians are better for angular speed because a radian is a ratio of two lengths.

9. Angular Speed Angle turned (rad) w = q t Angular speed (rad/sec) Time taken (sec)

10. Angular SpeedWhy does this formula work? Angle turned (rad) w = q t Angular speed (rad/sec) Time taken (sec) rad/s = rad s

11. Calculating angular speed • A bicycle wheel makes six turns in 2 seconds. • What is its angular speed in radians per second?

12. Calculating angular speed • Known: Unknown: 6 rotations w (rad/s) 2 s 1 rotation = 2π rad = 6 rot x 2 π rad 2 s 1 rot = 6 π rad/s w = q t

13. Linear and Angular Speed • A wheel rolling along the ground has both a linear speed and an angular speed. • A point at the edge of a wheel moves one circumference in each turn of the circle.

14. Linear and Angular Speed Radius (m) C = 2 Pr Circumference (m) Distance (m) 2 Pr v = d t Speed (m/sec) Time (sec)

15. Linear and Angular Speed Radius (m) v = w r Linear speed (m/sec) Angular speed (rad/sec) *This formula is used in automobile speedometers based on a tire's radius.

16. Linear and Angular SpeedWhy does this formula work? Circumference Radius (m) v = w r Linear speed (m/sec) w rad s m s 2πr m × 2π rad = Angular speed (rad/sec) v = r wm s Radians in one circumference

17. Calculating linear from angular speed • Siv is standing 4 meters from the axis of rotation and Holly is standing 2 meters from the axis. • Calculate each child’s linear speed when the angular speed of the merry go-round is 1 rad/sec. • Two children are spinning around on a merry-go-round.

18. Calculating linear from angular speed v1 = w r1 = (1)(2) = 2 m/s v2 = w r2 = (1)(4) = 4 m/s • Known: Unknown: • = 1 rad/s v1 r1 = 2 m v2 r2 = 4 m v = w r

19. Linear and Angular Speed and Displacement

20. Calculating angular from linear speed • The bicycle is moving forward with a linear speed of 11 m/sec. • Assume the bicycle wheels are not slipping and calculate the angular speed of the wheels in RPM (rotations per minute). • A bicycle has wheels that are 70 cm in diameter (35 cm radius).

21. Calculating angular from linear speed • Known: Unknown: v = 11 m/s w in rot/min r= 0.35 m 1 rot = 2π rad 1 min = 60 s 11 m/s = = 31.4 rad s 11 m/s__ 0.35 m/rad Circumference 31.4 rad x 1 rot x 60 s s 2π rad 1 min × w (0.35 m)(2π) 2 π rad = 300 rot/min = w Radians in one circumference

22. Key Question: Why does a roller coaster stay on a track upside down on a loop? Centripetal Force

23. Centripetal Force • We usually think of acceleration as a change in speed. • Because velocity includes both speed and direction, acceleration can also be a change in the direction of motion.

24. Centripetal Force • Any force that causes an object to move in a circle is called a centripetal force. • A centripetal force is always perpendicular to an object’s motion, toward the center of the circle.

25. Centripetal forces keep these children moving in a circular path.

26. v Constant velocity tangent to path. Constant force toward center. Uniform Circular Motion Uniform circular motion is motion along a circular path in which there is no change in speed, only a change in direction. Fc Question: Is there an outwardforce on the ball?

27. v Uniform Circular Motion (Cont.) The question of an outward force can be resolved by asking what happens when the string breaks! Ball moves tangent to path, NOT outward as might be expected. When central force is removed, ball continues in straight line. Centripetal force is needed to change direction.

28. Fc Examples of Centripetal Force You are sitting on the seat next to the outside door. What is the direction of the resultant force on you as you turn? Is it away from center or toward center of the turn? • Car going around a curve. Force ON you is toward the center.

29. Reaction The centripetal force is exerted BY the door ON you. (Centrally) Fc F’ Car Example Continued There is an outward force, but it does not act ON you. It is the reaction force exerted BY you ON the door. It affects only the door.

30. Fc n W v R Centripetal Acceleration Consider ball moving at constant speed v in a horizontal circle of radius R at end of string tied to peg on center of table. (Assume zero friction.) Force Fcand acceleration ac toward center. W = n

31. vf vf -vo vo vo R R Deriving Central Acceleration Consider initial velocity at A and final velocity at B: B Dv s A

32. Dv t Definition: ac = vf Similar Triangles -vo Dv s vo R mass m vs Rt Dv t vv R ac = = = Dv v s R = Deriving Acceleration (Cont.) Centripetal acceleration:

33. v R Car Negotiating a Flat Turn Fc What is the direction of the force ON the car? Ans.Toward Center This central force is exerted BY the road ON the car.

34. v Fc R Car Negotiating a Flat Turn Is there also an outward force acting ON the car? Ans.No, but the car does exert a outward reaction force ON the road.

35. n Fc R fs m R v mg The centripetal force Fc is that of static friction fs: Car Negotiating a Flat Turn Fc = fs The central force FC and the friction force fs are not two different forces that are equal. There is just one force on the car. The nature of this central force is static friction.

36. n Fc Fc = fs R fs m R v mv2 R Fc = mg Finding the maximum speed for negotiating a turn without slipping. The car is on the verge of slipping when FC is equal to the maximum force of static friction fs. fs = msmg Fc = fs

37. n fs R Fc R m v mg mv2 R = msmg v = msgR Maximum speed without slipping (Cont.) Fc = fs Velocity v is maximum speed for no slipping.

38. Centripetal Force Mass (kg) Linear speed (m/sec) Fc = mv2 r Centripetal force (N) Radius of path (m)

40. Calculating centripetal force • A 50-kilogram passenger on an amusement park ride stands with his back against the wall of a cylindrical room with radius of 3 m. • What is the centripetal force of the wall pressing into his back when the room spins and he is moving at 6 m/sec?

41. Centripetal Acceleration • Acceleration is the rate at which an object’s velocity changes as the result of a force. • Centripetal acceleration is the acceleration of an object moving in a circle due to the centripetal force.

42. Centripetal Acceleration Speed (m/sec) ac = v2 r Centripetal acceleration (m/sec2) Radius of path (m)

43. Calculating centripetal acceleration • A motorcycle drives around a bend with a 50-meter radius at 10 m/sec. • Find the motor cycle’s centripetal acceleration and compare it with g, the acceleration of gravity.

44. Centrifugal Force • We call an object’s tendency to resist a change in its motion its inertia. • An object moving in a circle is constantly changing its direction of motion. • Although the centripetal force pushes you toward the center of the circular path... • ...it seems as if there also is a force pushing you to the outside. This apparent outward force is called centrifugal force.

45. Centrifugal Force • Centrifugal force is not a true forceexerted on your body. • It is simply your tendency to move in a straight line due to inertia. • This is easy to observe by twirling a small object at the end of a string. • When the string is released, the object flies off in a straight line tangent to the circle.

47. Key Question: How strong is gravity in other places in the universe? Universal Gravitation and Orbital Motion

48. Universal Gravitation and Orbital Motion • Sir Isaac Newton first deduced that the force responsible for making objects fall on Earth is the same force that keeps the moon in orbit. • This idea is known as the law of universal gravitation. • Gravitational force exists between all objects that have mass. • The strength of the gravitational force depends on the mass of the objects and the distance between them.

49. Law of Universal Gravitation Mass 1 Mass 2 F = m1m2 r2 Force (N) Distance between masses (m)

50. Calculating gravitational force • The mass of the moon is 7.36 × 1022 kg. • The radius of the moon is 1.74 × 106 m. • Use the equation of universal gravitation to calculate the weight of a 90 kg astronaut on the surface of the moon.