# Physics. Chapter 8 Rotational Motion

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1 Physics Chapter 8 Rotational Motion

2 Circular Motion Tangential Speed The linear speed of something moving along a circular path. Symbol is the usual v and units are m/s Rotational Speed Number of revolutions per unit of time. Symbol is omega, ω and units are RPM (rotations per minute)

3 Circular Motion Does tangential speed depend on where the bug sits on the record? Does rotational speed depend on where the bug sits on the record?

4 Circular Motion Is there a relationship between v and ω? v ~ r ω Question: Who is moving at a greater tangential speed: someone at the equator or someone in Winchester, Virginia?

5 Circular Motion

6 Check Question At an amusement park, you and a friend sit on a large rotating disk. You sit at the edge and have a rotational speed of 4 RPM and a linear speed of 6 m/s. Your friend sits halfway to the center. What is her rotational speed? What is her linear speed?

7 Wheels on Trains

8 Period and Frequency All objects undergoing circular motion have one thing in common - they return to the same location every time they complete one cycle (one circumference). The time to complete one cycle is called the Period, T. T = t number of cycles ** Units seconds

9 Period and Frequency The frequency, f, of an object is the number of cycles an object completes during one second. In other words, how frequently, the object is cycling. Frequency is the inverse of Period. f = number of cycles 1 = t T ** Units cycles per second = Hertz

10 Check Question An object completes 20 revolutions in 10 seconds. Determine period and frequency of this motion.

11 Tangential Velocity Velocity involves both speed and direction. When an object moves in a circle, even at constant speed, the object still undergoes acceleration because its direction is changing. Several instantaneous velocity vectors are drawn in - notice they are all TANGENT to the circle.

12 Tangential Velocity We can calculate the magnitude of the tangential velocity as follows: d v = = t 2π r T

13 Centripetal Acceleration Since the direction of v is constantly changing for a circling object, we know it is continuously accelerating. This type of acceleration is called centripetal acceleration, a c. Centripetal acceleration is always directed toward the center of the circle.

14 Centripetal Acceleration We can calculate centripetal acceleration as follows: a c = v 2 r

15 Check Question Determine the acceleration of an object experiencing uniform circular motion. It is moving in a circle with a radius of 10 meters and a frequency of 0.25 Hertz.

16 Centripetal Force Recall that the cause of any acceleration is a force. In uniform circular motion, the net force is known as the centripetal force. From Newton s second law: F c = ma c = mv 2 r The centripetal force acts in the SAME direction as the centripetal acceleration.

17 Check Question - Horizontal Circular Motion A racing car is speeding around a flat, unbanked circular track whose radius is 250 meters. The car s speed is a constant 50.0 meters per second. The mass of the car is 2.00 x 103 kg. What provides the centripetal force necessary to keep the car in its circular path? Calculate the magnitude of the centripetal force in newtons on the car.

18 Centripetal Force Centripetal force holds a car in a curved path. a. For the car to go around a curve, there must be sufficient friction to provide the required centripetal force. b. If the force of friction is not great enough, skidding occurs.

19 Check Question - Horizontal Circular Motion A car moving at 10 meters / second completes a turn with a radius of 20 meters. Determine the minimum coefficient of friction between the tires and the road that will allow the car to complete the turn without skidding.

20 Check Question - Horizontal Circular Motion A mass m is in uniform circular motion with a speed v and a radius r. How is the centripetal acceleration affected if the radius is doubled while the speed remains constant?

21 Check Question - Vertical Circular Motion Imagine an ideal pendulum bob weighing 2.0 kg and swinging back and forth. At its lowest point (rest position) it has velocity 10 m/s. If the length of the pendulum string is 10 meters, what is the centripetal force on the pendulum bob at its lowest point?

22 Check Question - Vertical Circular Motion A 2.0-kilogram mass is attached to the end of a 1.0-meterlong string. When the apparatus is swung in a vertical circle, the mass reaches a speed of 10 meters per second at the bottom of the swing. A. Determine the tension in the string at the bottom of the swing. B. Determine the minimum speed at the top of the loop that allows the mass to make one complete cycle.

23 Check Question - Vertical Circular Motion A roller coaster loop has a radius of 10.0 meters. What minimum speed is required at the top of the loop in order to complete the loop successfully?

24 Centripetal Force

25 Calculating Centripetal Force c

26 Centripetal Force F v

27 Centripetal Force The clothes in a washing machine are forced into a circular path, but the water is not, and it flies off tangentially.

28 Centripetal Force T T y T x

29 Conical Pendulum

30 Banked Curves

31 Centrifugal Force

32 Centrifugal Force Effect The centrifugal-force effect is attributed not to any real force but to inertia the tendency of the moving body to follow a straight-line path.

33 Centrifugal vs Centripetal Force If the string on the can were to break, would the can move outward? No, because there is no outward force acting on it. It would move off in a straight line tangent to its location at the moment the string broke.

34 Check Question A heavy iron ball is attached by a spring to a rotating platform, as shown in the sketch. Two observers, one in the rotating frame and one on the ground at rest, observe its motion. Which observer sees the ball being pulled outward, stretching the spring? Which observer sees the spring pulling the ball into circular motion?

35 Check Question Mass m is positioned 1.0 meter from the center of a circular disk that is rotating with increasing speed, as shown bleow. The coefficient of friction between the mass and the disk is What maximum speed can the mass obtain before it slips off the rotating disk?

36 Simulated Gravity From within a rotating frame of reference, there seems to be an outwardly directed centrifugal force, which can simulate gravity.

37 Simulated Gravity The man inside this rotating space habitat experiences simulated gravity. As seen from the outside, the only force exerted on the man is by the floor. As seen from the inside, there is a fictitious centrifugal force that simulates gravity.

38 Challenges of Simulated Gravity The magnitude of the centripetal/centrifugal force is directly proportional to the radial distance. Human sensitivity to rotation. Large structure required

39 Rotational Inertia An object rotating about an axis tends to remain rotating about the same axis unless interfered with by some external influence. The greater the distance between an object s mass concentration and the axis, the greater the rotational inertia.

40 Rotational Inertia

41 Rotational Inertia So how can we calculate rotational inertia? I = mr 2 ONLY for the case where all the mass, m, is concentrated at the same distance, r, from a rotational axis. When the mass is more spread out, the rotational inertia is less and the formula is different.

42 Rotational Inertia

43 Rotational Inertia Which will roll down an incline with greater acceleration, a hollow cylinder or a solid cylinder of the same mass and radius?

44 Rotational Inertia It turns out that any solid cylinder will roll down an incline with more acceleration than any hollow cylinder, regardless of mass or radius. A hollow cylinder has more laziness per mass than a solid cylinder.

45 Check Question A heavy iron cylinder and a light wooden cylinder, similar in shape, roll down an incline. Which will have more acceleration? The cylinders have different masses, but the same rotational inertia per mass, so both will accelerate equally down the incline. Their different masses make no difference, just as the acceleration of free fall is not affected by different masses. All objects of the same shape have the same laziness per mass ratio.

46 Rotational Inertia Rotational inertia can be used to advantage. Industrial flywheels

47 Torque Torque is the rotational counterpart to Force Apply a force if you want linear motion (acceleration) Apply a torque if you want rotation.

48 Torque Now we have to expand our definition of Mechanical Equilibrium! F = 0 AND τ = 0

49 Torque Let s consider a door:

50 Torque When the force is perpendicular, the distance from the turning axis to the point of contact is called the lever arm.

51 Torque The same torque can be produced by a large force with a short lever arm, or a small force with a long lever arm.

52 Check Question If you cannot exert enough torque to turn a stubborn bolt, would more torque be produced if you fastened a length of rope to the wrench handle as shown?

53 Check Question

54 Balanced Torques A pair of torques can balance each other. Balance is achieved if the torque that tends to produce clockwise rotation by the boy equals the torque that tends to produce counterclockwise rotation by the girl.

55 Check Question This meter stick is suspended in mechanical equilibrium. What is the weight of the block hanging at the 10 cm mark?

56 Balanced Torques Scale balances that work with sliding weights are based on balanced torques, not balanced masses. The sliding weights are adjusted until the counterclockwise torque just balances the clockwise torque. We say the scale is in rotational equilibrium.

57 Center of Mass A baseball thrown into the air follows a smooth parabolic path.

58 Center of Mass A bat thrown in the air, however, wobbles about a special point. This point stays on a parabolic path, even though the rest of the bat does not. This point, called the center of mass, is where all the mass of an object can be considered to be concentrated.

59 Center of Mass Center of Mass is the average position of all mass that makes up the object Center of Gravity is the average position of all the particles of weight that make up an object

60 Center of Mass Location of the Center of Mass

61 Center of Mass Objects not made of the same material throughout may have the center of mass quite far from the geometric center. Consider a hollow ball half filled with lead. The center of mass would be located somewhere within the lead part. The ball will always roll to a stop with its center of mass as low as possible.

62 Center of Mass Motion About the Center of Mass As an object slides across a surface, its center of mass follows a straight-line path.

63 Spin A force must be applied to the edge of an object for it to spin. a. If the football is kicked in line with its center, it will move without rotating. b. If it is kicked above or below its center, it will rotate.

64 Center of Gravity So how can we precisely locate the center of gravity (CG)? The CG of a uniform object is at the midpoint, its geometric center. The CG is the balance point. Supporting that single point supports the whole object.

65 Center of Gravity The CG of an object may be located where no actual material exists. The CG of a ring lies at the geometric center where no matter exists. The same holds true for a hollow sphere such as a basketball.

66 Check Question A Uniform meterstick supported at the 25-cm mark balances when a 1 kg rock is suspended at the 0-cm end. What is the mass of the meterstick?

67 Torque & Center of Gravity

68 Torque & Center of Gravity

69 Torque & Center of Gravity The Leaning Tower of Pisa does not topple because its CG does not extend beyond its base.

70 Center of Gravity

71 Balancing Question: Which way do you suppose it will be easier to balance a baseball bat on your finger?

72 Stability The CG of a building is lowered if much of the structure is below ground level. This is important for tall, narrow structures.

73 Spools

74 Angular Momentum Anything that rotates keeps on rotating until something stops it. Recall what we learned about linear momentum: P = linear inertia x linear velocity = m x v Angular momentum is defined as the product of rotational inertia, I, and rotational velocity, ω. angular momentum = rotational inertia rotational velocity

75 Angular Momentum Angular momentum depends on rotational velocity and rotational inertia. The operation of a gyroscope relies on the vector nature of angular momentum.

76 Angular Momentum Just as an external net force is required to change the linear momentum of an object. An external net torque is required to change the angular momentum of an object. An object or system of objects will maintain its angular momentum unless acted upon by an external net torque.

77 Conservation of Angular Momentum Angular momentum is conserved for systems in rotation. The law of conservation of angular momentum states that if no unbalanced external torque acts on a rotating system, the angular momentum of that system is constant. With no external torque, the product of rotational inertia and rotational velocity at one time will be the same as at any other time.

78 Conservation of Angular Momentum When the man pulls his arms and the whirling weights inward, he decreases his rotational inertia, and his rotational speed correspondingly increases.

79 Conservation of Angular Momentum Rotational speed is controlled by variations in the body s rotational inertia as angular momentum is conserved during a forward somersault. This is done by moving some part of the body toward or away from the axis of rotation.

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