# Rotational Mechanics - 1

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1 Rotational Mechanics - 1

2 The Radian The radian is a unit of angular measure. The radian can be defined as the arc length s along a circle divided by the radius r. s r

4 Average Angular Speed The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval av f i t t t f i

5 Instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero lim t 0 t Units of angular speed are radians/sec (rad/s). Speed is positive if θ is increasing (counterclockwise rotation). Speed is negative if θ is decreasing (clockwise rotation).

6 A particle moves along a circular path of radius R with velocity of constant magnitude u. The time for one rotation is the period, T.

7 Basic Definitions & Units Period T Frequency : time for one rotation f : # of rotations/time interval 1 f If T is in seconds s, f is in s T Angular velocity magnitude : 2 f Unit for angular velocity: radians sec -1 2 T

8 The magnitude of the particle's velocity is the ratio of distance moved in one rotation to the time (period) required for one rotation. u 2 R T

9 The magnitude of the particle's velocity is the ratio of distance moved in one rotation to the time (period) required for one rotation. 2 R u T This can also be expressed in terms of the frequency f : u 2 Rf

10 The magnitude of the particle's velocity is the ratio of distance moved in one rotation to the time (period) required for one rotation. 2 R u T This can also be expressed in terms of the frequency f : u 2 Rf and in terms of the angular velocity u R

11 In time interval t the particle moves along an arc of length s. The magnitude of the particle s velocity is s/ t.

12 The CHANGE in the velocity vector during this time interval is u.

13 The CHANGE in the velocity vector during this time interval is u.

14 The CHANGE in the velocity vector during this time interval is u. The particle moves a horizontal distance x during this time interval.

15 Over a shorter time interval, the length of the arc s approaches the length of the chord x. The velocity CHANGE is u = u 2 + (-u 1 )

16 These triangles are similar, i.e., their sides have the same ratio: u x u or u x u R R and as the angle gets smaller s approaches x in length.

17 These triangles are similar, i.e., their sides have the same ratio: u x u or u x u R R and as the angle gets smaller s approaches x in length, so u u s. R

18 The magnitude of the average acceleration is then u u s u u aav u. t R t R R This is the centripetal acceleration of the particle.

19 Centripetal acceleration can be expressed in these equivalent forms: u 2 4 R a c R 4 Rf 2 R T 2 2

20 Average and Instantaneous Angular Acceleration Average angular acceleration a av of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change: f i aav t t t f i Instantaneous angular acceleration: a lim t 0 t

21 Relating Linear and Angular Motion If u is constant: If is constant: x ut t

22 Relating Linear and Angular Motion If a is constant: If a is constant: 1 1 x u t at t at i 2 2 i 2 2 i u u at at u u 2a x 2a i s r Tangential velocity: u Tangential acceleration: t r a t ra

23 Total Acceleration The tangential component of the acceleration is due to changing speed. The centripetal component of the acceleration is due to changing direction. Total acceleration can be found from these components 2 2 t c a a a

24 Grasp the axis of rotation with your right hand. Wrap your fingers in the direction of rotation. Your thumb points in the direction of ω. Vector Nature of Angular Quantities Angular displacement, velocity and acceleration are all vector quantities. Direction can be more completely defined by using the right hand rule

25 Angular Velocity Directions, Example In (a), the disk rotates clockwise. The angular velocity is into the page/screen. In (b), the disk rotates counterclockwise. The angular velocity is out of the page/screen.

26 Centripetal Acceleration and Centripetal Force From Newton s second law: F c = ma c Centripetal force, F c, is not an added force. F c can be supplied by a force that maintains an object on a circular path. Examples: - String tension - Gravity - Friction

27 Centripetal Force Example 1 A ball of mass m is attached to a string. Its weight is supported by a frictionless table. The tension in the string causes the ball to move in a circle. The centripetal force is supplied by the string tension: F ma T c c mu r 2

28 Centripetal Force Example 2 Car on a level curve The centripetal force is supplied by the static friction force. mu 2 F f mg C s s r u s gr

29 Centripetal Force Example 3 Car on a banked curve The centripetal force is supplied by the horizontal component of the normal force. 2 mu nsin n cos mg r 2 nsin u tan n cos gr a c gtan

30 Rotational Mechanics Questions

31 A particle rotating in a circle undergoes uniform angular acceleration. Which (one or more) of these graphs could be associated with this kind of motion?

32 A particle rotating in a circle undergoes uniform angular acceleration. Which (one or more) of these graphs could be associated with this kind of motion? A, C

33 A particle rotating in a circle undergoes uniform angular acceleration. Which (one or more) of these graphs could be associated with this kind of motion?

34 A particle rotating in a circle undergoes uniform angular acceleration. Which (one or more) of these graphs could be associated with this kind of motion? A, B, C, E

35 A particle rotating in a circle undergoes uniform angular acceleration. Which (one or more) of these graphs could be associated with this kind of motion?

36 A particle rotating in a circle undergoes uniform angular acceleration. Which (one or more) of these graphs could be associated with this kind of motion? A, B, C, D

37 A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center of the disk as point P is. The angular velocity of Q at a given time is A. twice as big as P's. B. the same as P's. C. half as big as P's. D. none of the above.

38 A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center of the disk as point P is. The angular velocity of Q at a given time is A. twice as big as P's. B. the same as P's. C. half as big as P's. D. none of the above.

39 The figure shows the velocity and acceleration of a particle at a particular instant in three situations In which situation, and at that instant, is the speed increasing?

40 The figure shows the velocity and acceleration of a particle at a particular instant in three situations In which situation, and at that instant, is the speed increasing? B

41 The figure shows the velocity and acceleration of a particle at a particular instant in three situations In which situation, and at that instant, is the speed decreasing?

42 The figure shows the velocity and acceleration of a particle at a particular instant in three situations In which situation, and at that instant, is the speed increasing? C

43 The figure shows the velocity and acceleration of a particle at a particular instant in three situations In which situation, and at that instant, is the speed not changing?

44 The figure shows the velocity and acceleration of a particle at a particular instant in three situations In which situation, and at that instant, is the speed not changing? A (a is centripetal)

45 A particle P moves in a circle in a horizontal plane. The motion is counterclockwise, as viewed from above. If the particle undergoes positive angular acceleration, which drawing correctly represents the net horizontal force (red arrow) applied to the particle?

46 A particle P moves in a circle in a horizontal plane. The motion is counterclockwise, as viewed from above. If the particle undergoes positive angular acceleration, which drawing correctly represents the net horizontal force (red arrow) applied to the particle? C

47 If the particle undergoes positive angular acceleration, which drawing correctly represents the net horizontal force (red arrow) applied to the particle? C (The net force is the vector sum of the tangential and centripetal forces.)

48 If the particle moves with constant angular velocity, which drawing correctly represents the net horizontal force (red arrow) applied to the particle?

49 If the particle moves with constant angular velocity, which drawing correctly represents the net horizontal force (red arrow) applied to the particle? A

50 A nut is located m from the axis of a machine part that rotates uniformly at a rate of 120. times per second. Calculate the period of rotation.

51 A nut is located m from the axis of a machine part that rotates uniformly at a rate of 120. times per second. Calculate the period of rotation. T = (1/120) s

52 A nut is located m from the axis of a machine part that rotates uniformly at a rate of 120. times per second. Calculate the distance the nut moves in 2.00 seconds.

53 A nut is located m from the axis of a machine part that rotates uniformly at a rate of 120. times per second. Calculate the distance the nut moves in 2.00 seconds. 2 r T s t 2 r t T 2.25 m 2 s 120 m 377 m 1120 s

54 Calculate the time required for the nut to move through an angle of π radians.

55 Calculate the time required for the nut to move through an angle of π radians. t 2 T t t T T T 1120 s 1 s

56 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole.

57 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole. Write down everything you know:

58 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole. Write down everything you know: M = 55 kg, u = 4 m/s, R =.8 m

59 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole. (a) Determine the force exerted by the horizontal rope on her arms.

60 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole. (a) Determine the force exerted by the horizontal rope on her arms. This force is A. tangent to the circular path. B. directed inward. C. directed outward.

61 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole. (a) Determine the force exerted by the horizontal rope on her arms. This force is A. tangent to the circular path. B. directed inward (centripetal force). C. directed outward.

62 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole. (a) Determine the force exerted by the horizontal rope on her arms. F c 2 2 mu 55 kg 4 m s R.8 m 1100 N

63 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole. (b) Compare this force (1100 N) with her weight.

64 7.21 A 55.0-kg ice skater is moving at 4.00 m/s when she grabs the loose end of a rope, the opposite end of which is tied to a pole. She then moves in a circle of radius m around the pole. (b) Compare this force (1100 N) with her weight. F c mg 1100 N 2 55 kg 9.8 m s

65 7.23 A truck can go around a flat curve of radius 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve of 75 m?

66 7.23 A truck can go around a flat curve of radius 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve of 75 m? The centripetal force is supplied by A. Gravity B. The car s brakes C. Static friction

67 7.23 A truck can go around a flat curve of radius 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve of 75 m? The centripetal force is supplied by A. Gravity B. The car s brakes C. Static friction

68 7.23 A truck can go around a flat curve of radius 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve of 75 m? R 150 m, 32 m s, R 75 m 1 1 2

69 7.23 A truck can go around a flat curve of radius 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve of 75 m? Assumption:

70 7.23 A truck can go around a flat curve of radius 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve of 75 m? Assumption: Centripetal force (static friction force) does not change.

71 7.23 A truck can go around a flat curve of radius 150 m with a maximum speed of 32.0 m/s. With what maximum speed can it go around a curve of 75 m? Equate expressions for centripetal acceleration: R 2 R 1 2 R 75 m 32 m s 22.6 m s R1 150 m

72 7.24 A sample of blood is in a centrifuge of radius 15 cm. The mass of a red blood cell is 3.0 x kg and the magnitude of the force acting on it is 4.0 x N. At how many revolutions per second should the centrifuge be operated?

73 7.24 A sample of blood is in a centrifuge of radius 15 cm. The mass of a red blood cell is 3.0 x kg and the magnitude of the force acting on it is 4.0 x N. At how many revolutions per second should the centrifuge be operated? The centripetal force is supplied by A. Gravity B. Friction C. The bottom of the blood sample container.

74 7.24 A sample of blood is in a centrifuge of radius 15 cm. The mass of a red blood cell is 3.0 x kg and the magnitude of the force acting on it is 4.0 x N. At how many revolutions per second should the centrifuge be operated? The centripetal force is supplied by A. Gravity B. Friction C. The bottom of the blood sample container.

75 7.24 A sample of blood is in a centrifuge of radius 15 cm. The mass of a red blood cell is 3.0 x kg and the magnitude of the force acting on it is 4.0 x N. At how many revolutions per second should the centrifuge be operated? Given: R =.15 m, m = 3 x kg, F c = 4 x N

76 7.24 A sample of blood is in a centrifuge of radius 15 cm. The mass of a red blood cell is 3.0 x kg and the magnitude of the force acting on it is 4.0 x N. At how many revolutions per second should the centrifuge be operated? F f c 4 Rf m 2 2 F 11 c 4 10 N Rm 4.15 m 3 10 kg s rev s

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