Chapter 8: Newton s Laws Applied to Circular Motion

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1 Chapter 8: Newton s Laws Applied to Circular Motion

2 Centrifugal Force is Fictitious? F actual = Centripetal Force F fictitious = Centrifugal Force Center FLEEing

3 Centrifugal Force is Fictitious? Center FLEEing Suppose there is a lady bug in the can. There is a centripetal force acting on the bug, transmitted to her feet by the can. Her feet push back on the can producing a centrifugal fictitious force, that acts like artificial gravity.

4 Centripetal & Centrifugal Force Depends on Your Reference Frame Center-seeking Outside Observer (non-rotating frame) sees Centripetal Force pulling can in a circle. Center-fleeing Inside Observer (rotating reference frame) feels Centrifugal Force pushing them against the can.

5 Centrifugal Force is Fictitious? The centrifugal force is a real effect. Objects in a rotating frame feel a centrifugal force acting on them, trying to push them out. This is due to your inertia the fact that your mass does not want to go in a circle. The centrifugal force is called fictitious because it isn t due to any real force it is only due to the fact that you are rotating. The centripetal force is real because it is due to something acting on you like a string or a car.

6 The Earth rotates once per day around its axis as shown. Assuming the Earth is a sphere, is the rotational speed at Santa Rosa greater or less than the speed at the equator? 366 m/s 464 m/s

7 What is the total acceleration acting on a person in Santa Rosa? The vector sum. a c g

8 Is your apparent weight as measured on a spring scale more at the Equator or at Santa Rosa? a c g

9 Since you are standing on the Earth (and not in the can) the centrifugal force tends to throw you off the Earth. You weigh less where the centripetal force is greatest because that is also where the centrifugal force is greatest the force that tends to throw you out of a rotating reference frame.

10 Artificial Gravity How fast would the space station segments A and B have to rotate in order to produce an artificial gravity of 1 g? v 56 m/ s ~115mph A v 104 m/ s ~ 210mph B Can the two segments be connected?

11 Coriolis Force This is an apparent force caused by changing the radial position of an object in a rotating coordinate system The result of the rotation is the curved path of the ball

12 Coriolis Force This is an apparent force caused by changing the radial position of an object in a rotating coordinate system The result of the rotation is the curved path of the ball

13 Coriolis Effect

14

15 Translational and Rotational Kinematics For CONSTANT Accelerations ONLY

16 Total Acceleration & Force a t d dt v 2 a a a a r a 2 2 r t C v r F F 2 F 2 r

17 Horizontal Circle: Constant Speed & Acceleration Vertical Circle: Changing Speed & Acceleration

18 Important: Inside vs Outside the Rotating Frame

19 Motion in a Horizontal Circle The speed at which the object moves depends on the mass of the object and the tension in the cord. It is constant! The centripetal force is supplied by the tension. Looking down: c F T ma c mv r 2 v Tr m

20 Motion in a Horizontal Circle

21 Horizontal (Flat) Curve The force of static friction supplies the centripetal force Fc The maximum speed at which the car can negotiate the curve is f mv r 2 v gr Note, this does not depend on the mass of the car y F N mg 0 f mg

22 Horizontal (Flat) Curve 3. A highway curve has a radius of 0.14 km and is unbanked. A car weighing 12 kn goes around the curve at a speed of 24 m/s without slipping. What is the magnitude of the horizontal force of the road on the car? What is μ? Draw FBD. a. 12 kn b. 17 kn c. 13 kn d. 5.0 kn e. 49 kn

23 Banked Curve These are designed with friction equaling zero - there is a component of the normal force that supplies the centripetal force that keeps the car moving in a circle. Fr y 2 mv nsin r F ncos mg 0 Dividing: tan 2 v rg

24 Banked Curve A race car travels 40 m/s around a banked (45 with the horizontal) circular (radius = 0.20 km) track. What is the magnitude of the resultant force on the 80-kg driver of this car? a kn b kn c kn d kn e kn

25 Hints for HW Problem Determine the range of speeds a car can have without slipping up or down the road when it is banked AND has friciton. If the car is about to slip down the incline, f is directed up the incline. This would happen at a minimum speed. When the car is about to slip up the incline, f is directed down the incline. This would happen at a maximum speed.

26 Vertical Circle with Non-Uniform Speed Where is the speed Max? Min? Where is the Tension Max? Min?

27 Vertical Circle with Non-Uniform Speed The tension at the bottom is a maximum The tension at the top is a minimum Look at radial and tangential: t F mg sin ma t at gsin r F T mg cos mv R 2 2 v T m g R cos

28 Vertical Circle: Mass on a String A 0.40-kg mass attached to the end of a string swings in a vertical circle having a radius of 1.8 m. At an instant when the string makes an angle of 40 degrees below the horizontal, the speed of the mass is 5.0 m/s. What is the magnitude of the tension in the string at this instant? Draw the FBD. a. 9.5 N b. 3.0 N c. 8.1 N d. 5.6 N e. 4.7 N

29 Vertical Circle: Mass on a String A 0.30-kg mass attached to the end of a string swings in a vertical circle (R = 1.6 m), as shown. At an instant when = 50, the tension in the string is 8.0 N. What is the magnitude of the total force on the mass at this instant? a. 5.6 N b. 6.0 N c. 6.5 N d. 5.1 N e. 2.2 N 2 2 Hint: F Fr F

30 Minimum Speed for Vertical Circular Motion What is the minimum speed so that the ball can go in the circle? That is, when T = 0 at the top? At the top: v T m g cos 0 R v gr Minimal Speed to JUST get around the circle only depends on R! ROOT GRRRRRRRR

31 QuickCheck 8.11 Loop d Loops: Inside the Vertical Loop A roller coaster car does a loopthe-loop. Which of the free-body diagrams shows the forces on the car at the top of the loop? Rolling friction can be neglected. Slide 8-82

32 QuickCheck 8.11 Loop d Loops: Inside the Vertical Loop A roller coaster car does a loopthe-loop. Which of the free-body diagrams shows the forces on the car at the top of the loop? Rolling friction can be neglected. The track is above the car, so the normal force of the track pushes down. Slide 8-83

33 Loop d Loops: Inside the Vertical Loop A roller-coaster car has a mass of 500 kg when fully loaded with passengers. At the bottom of a circular dip of radius 40 m (as shown in the figure) the car has a speed of 16 m/s. What is the magnitude of the force of the track on the car at the bottom of the dip? a. 3.2 kn b. 8.1 kn c. 4.9 kn d. 1.7 kn e. 5.3 kn

34 Loop d Loops: Inside the Vertical Loop Minimum Speed to get to the Top. What is the minimum speed so that the car barely make it around the loop the riders are upside down and feel weightless? R = 10.0m

35 QuickCheck 8.10 Humps in the Road: Outside the Vertical Loop A car that s out of gas coasts over the top of a hill at a steady 20 m/s. Assume air resistance is negligible. Which free-body diagram describes the car at this instant? Slide 8-80

36 QuickCheck 8.10 Humps in the Road: Outside the Vertical Loop A car that s out of gas coasts over the top of a hill at a steady 20 m/s. Assume air resistance is negligible. Which free-body diagram describes the car at this instant? Now the centripetal acceleration points down. Slide 8-81

37 Humps in the Road Outside the Vertical Loop A roller-coaster car has a mass of 500 kg when fully loaded with passengers. The car passes over a hill of radius 15 m, as shown. At the top of the hill, the car has a speed of 8.0 m/s. What is the force of the track on the car at the top of the hill? a. 7.0 kn up b. 7.0 kn down c. 2.8 kn down d. 2.8 kn up e. 5.6 kn down

38 Maximum Speed for Vertical Circular Motion : Humps in the Road What is the maximum speed the car can have as it passes this highest point without losing contact with the road? Max speed without losing contact MEANS: n Take : n 0 Therefore: mg mv r 2 v gr mg Maximum Speed to not loose contact with road only depends on R! ROOT GRRRRRRRR

39 What is the maximum speed the vehicle can have at B and still remain on the track?

40 Hump in the Road Suppose that a kg car passes over a bump in a roadway that follows the arc of a circle of radius 20.4 m. (a) What force does the road exert on the car as the car passes the highest point of the bump if the car travels at 30.0 km/h? (b) What If? What is the maximum speed the car can have as it passes this highest point without losing contact with the road?

41 Chapter 6 Problem#51 v : 1 h 1000 m 30 km h 8.33 m s 3600 s 1 km F y n m g 4 m a y 2 mv r 2 v 2 n m g 1800 kg 9.8 m s r N up Suppose that a kg car passes over a bump in a roadway that follows the arc of a circle of radius 20.4 m. a) What force does the road exert on the car as the car passes the highest point of the bump if the car travels at 30.0 km/h? Minus because ay is pointing down m s m n mg +

42 Chapter 6 Problem#51 Suppose that a kg car passes over a bump in a roadway that follows the arc of a circle of radius 20.4 m. (b) What If? What is the maximum speed the car can have as it passes this highest point without losing contact with the road? : F y mg m a n m g y 2 mv 2 mv r r 2 Take : n 0 n mg v gr 9.8 m s 20.4 m 14.1 m s 50.9 km h

43 Vertical Motion: Constant Speed. Loop-the-Loop: UCM This is an example of a vertical circle with constant motion. The constant motion is maintained by an engine. Where is the force on the pilot the greatest, at the top or the bottom of circle? Is the force greater or less than her weight?

44 Vertical Motion: Constant Speed. Loop-the-Loop: UCM This is an example of a vertical circle with constant motion. The constant motion is maintained by an engine. At the bottom of the loop, the upward force experienced by the pilot is greater than at the top and is greater that her weight: y v F n mg m bot r 2 n bot mg 1 2 v rg

45 Vertical Motion: Constant Speed. Loop-the-Loop: UCM At the top of the circle, the force exerted on the pilot is less than her weight: y v F n mg m bot r 2 v 2 n mg top rg 1

46 Vertical Motion: Constant Speed. Loop-the-Loop: UCM An airplane moves 140 m/s as it travels around a vertical circular loop which has a 1.0-km radius. What is the magnitude of the resultant force on the 70-kg pilot of this plane at the bottom of this loop? a. 2.1 kn b. 1.4 kn c kn d. 1.5 kn e. 1.3 kn

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