Name Period Date A) B) C) D)

Transcription

1 Example Problems 9.2 E1. A car rounds a curve of constant radius at a constant speed. Which diagram best represents the directions of both the car s velocity and acceleration? Explain: A) B) C) D) E2. A ball attached to a string is moves in a clockwise circular path as seen from above. Circle the point that the string should be released so that the ball hits the target. E3. A monkey is driving around a corner that has a radius of 25 meters at a speed of 8.8 m/s. Calculate is the centripetal acceleration acting on the car. a c = 1

2 E4. A dog is chasing his tail and runs in a circle of radius 0.62 meters. The dog runs in a circle 10 times in 7.2 seconds. Calculate the centripetal acceleration of the dog a c = b) E5. A popular trick is to swing a pail of water around in a vertical circle fast enough so that the water doesn t spill out when the pail is upside down. The person swinging the pain has an arm length of 0.87 meters long, and the level of the water is 0.25 meters below the handle of the pail. Calculate the minimum speed with which he can swing the pail so that the water doesn t spill out at the top of the path. a c = 2

3 E6. A 40.0-kg stone is whirled around horizontally on the end of a 0.60 meters string at 2.2 m/s. Calculate the centripetal acceleration and the tension in the string. F c = m = a c = b) E7. An athlete whirls a 7.00-kg hammer tied to the end of a 1.3-m chain in a horizontal circle. The tension in the chain is 357 N, calculate the tangential speed of the hammer. F c = m = 3

4 E8. A monkey is doing his laundry. A sock is stuck to the inside of the clothes dryer spins around the drum once every 2.0 seconds at a distance of 0.50 meters from the center of the drum, calculate the sock s linear speed. T = b) E9. A monkey is riding on a merry-go-round sitting on a horse that is 8.0 m from the center of the ride. If the monkey has a mass of 55 kg and the merry-go-round turns around once every 40.0 sec, what is the centripetal acceleration and centripetal force? F c = m = a c = b) 4

5 E10. Let the block in Figure represent a Cadillac, mass 1610 kg, is moving at a constant speed of 20 m/s on an unbanked curved roadway whose radius of curvature is 190 meters. What must be the minimum coefficient of friction between the tires and the roadway be so that the car makes the turn? E11. You cannot count on a sideways frictional force to get your car around a curve if the road is icy or wet. That is why highways are banked. As in the pervious problem, suppose that the car of mass 1610 kg is moving at a speed of 20 m/s around a curve whose radius is 190 m. What angle of banking would make reliance on friction unnecessary? 5

6 E12. A mass 0.2 kg slides on a frictionless track with a circular loop, as shown. The mass is released from rest from a height of 1 m and the diameter for the loop is 0.3 m. Calculate the force the track exerts on m at points A, B, and C. E13. A conical pendulum s bob has a mass of 1.5 kg. It whirls around in a horizontal circle at constant speed at the end of a cord whose length, measured to the center of the bob, is 1.7 m. The cord makes an angle q of 37 with the vertical. As the bob swings around in a circle, the cord sweeps out the surface of a cone. Find the period of the pendulum, that is, the time for the bob to complete one circle. 6

7 Student Problems A car travels at a constant speed of 20 m/s around a horizontal circular track. Which diagram correctly represents the direction of the car s velocity (v) and the direction of the centripetal force (Fc) acting on the car at one particular moment? v F c F c v v F c v F c Explain: ( 1 ) ( 2 ) ( 3 ) ( 4 ) 2. During an Olympic bobsled run, a Planet of the Apes team takes a turn of radius 7.62 meters at a speed of 60 mph (26.82 m/s). Calculate the centripetal acceleration the team members experience during the turn. a c = 7

8 3. A 2,000-kilogram car is traveling at a constant speed of 12 m/s as it rounds a circular curve of radius 30 meters. What is the magnitude of the centripetal force and the centripetal acceleration of the car as it goes around the curve? F c = m = a c = b) 4. A plane is piloted by a monkey and is traveling at 224 m/s. The monkey makes a sharp turn following a circular trajectory, what should the radius of the trajectory be so that the acceleration acting on the monkey does not exceed 4 times the acceleration of gravity? [use 9.8 as gravity] a c = 8

9 5. A monkey is driving out of control when he enters into a turn traveling way to fast. The car has a mass of 2,000 kg and is traveling at a speed of 80 mph (35.76 m/s) as it enters the turn. If the total amount of friction (centripetal force) possible between the tires and the road is only 9,790 N, what is the minimum radius possible so that the car can make without flying off the road? If the actual radius of the turn is only 120 meters from the center does the monkey make it through the turn? F c = m = 6. A force of 150 N is required to break string. A 1.2 kg mass is fixed to one end of the cord and whirled around horizontally at a 3 meter radius. Calculate the maximum linear velocity of the mass so that the string does not beak. F c = m = b) 9

10 7. A monkey s favorite ride at the fair is the Ferris wheel that has a radius of 7.0 m. If the monkey has a mass of 50 kg, how fast must the Ferris wheel be turning so that the monkey feels weightless at the top? a c = 8. In a 1901 circus performance, Allo Dare Devil Diavolo introduced the stunt of riding a bicycle in a loop-the-loop. Assuming the loop is a circle with radius 2.7 meters, what is the least linear velocity he could have at the top of the loop if he is to remain in contact with the track at the top? a c = 10

11 9. A front loading clothes washer has a horizontal drum that is thoroughly perforated with small holes. A monkey is washing his 4.5 kg Teddy bear. Assuming the washer spin at 1 rotation per second and has a radius of 0.40 meters, what is the linear velocity of the Teddy bear and the centripetal force exerted on the Teddy bear? F c = m = b) 10. Captain Chip, a monkey, pilots a 60,500-kg jet plane. Chip is told that he must remain in a holding pattern over the airport until it is his turn to land. If Chip flies his plane in a circle whose radius is 500,000 meters once every 30.0 min, calculate the linear velocity of the plane and the centripetal force acting on the plane. F c = m = b) 11

12 11. At an amusement park, a monkey whose mass is 65 kilograms rides in a cage that completes one rotation every 8 seconds. The cage travels in a vertical circular path of radius 15 meters. Calculate the magnitude of the centripetal acceleration and the centripetal force acting on the monkey. F c = m = a = c b) 12

13 Monkey Busters Two masses,1 kg and 0.5 kg, are connected by a massless string that passes through a hole of a horizontal friction-free table. The 1-kg mass is suspended below the table and is in equilibrium when the other mass moves in a circular path of 0.20 meter radius on the table. Calculate the linear velocity of the 0.5-kg mass on the table. 13. A small coin is placed on a record that is rotating at 33.3 rpm. If the coefficient of static friction between record and coin is 0.12, how far from the center of the record can one place the coin without having it slip off? 13

14 14. What is the smallest radius of an unbanked curve around which a bicyclist can travel if her speed is 18 mph and the coefficient of static friction between the tires and the road is 0.32? 15. An unbanked curve has a radius of curvature of 130 meters. If the coefficient of static friction between a car s tires and the pavement is 0.6, what is the maximum speed at which the car can negotiate this curve without skidding? 14

15 16. A car enters an unbanked curve of 100 meters radius at a speed of 100 km/h. If the car does not skid, what is the minimum coefficient of static friction between tires and pavement? 17. A monkey, 50.0-kg swings from the end of a 5.0-meter long vine attached to a tree branch. The tree branch will break if subjected to a force greater than 750 N. What is the maximum speed with which the monkey can swing in order to avoid breaking the branch? b) 15

16 18. A stunt pilot is diving toward the earth at an angle of 45, traveling at a speed of 1200 km/h. At an altitude of 800 meters, he pulls back on the stick and puts the plane into a circular turn in the vertical plane. What is the maximum acceleration to which his body will be subjected if he is to avoid crashing the plane? 19. At a given instant, someone strapped into a roller coaster car hangs upside down at the very top of the circle (of radius 25.0 meters) while executing a so called loop the loop. At what speed must he be traveling if at that moment the force exerted by his body on the seat is half his actual weight? 16

17 20. The radius of curvature of a loop-the-loop roller coaster is 6 meters. If at the top of the loop the force which the seat exerts on the passenger is 0.5 mg, directed downward, what is then the speed of the roller coaster? From what initial elevation should the roller coaster start (assuming that it starts from rest) to achieve this speed? Neglect friction losses. b) 21. A 1000 kg car traveling on a road that runs straight up a hill reaches the rounded crest at 10.0 m/s. If the hill at that point has a radius of curvature (in a vertical plane) of 50 meters, what is the effective weight of the car at the instant it is horizontal at the very peak? 17

18 22. At the two ends of the oval of an indoor bicycle track, the radius of curvature of the track is 12 meters. What should the banking angle be if the normal racing speed averages 12.5 m/s? 23. What is the correct banking angle for a curve of 300 meter radius for a car traveling at a speed of 60 km/h assuming no force of friction? 18

19 24. A curve whose radius is 300 m is banked at an angle of 15. What is the optimum speed for negotiating this curve? 25. A curve of radius 120 meters is banked at an angle of 12. If a car of mass 1000 kg travels around this curve at m/s without slipping what is the frictional force between car tires and pavement? 19

20 26. Repeat the previous problem for a car of the same mass, traveling around the curve at 25 m/s. What is the minimum value of the coefficient of static friction that will prevent skidding? 27. An 0.8-kg pendulum bob is supported by a 1.5-m-long string from a fixed point. The bob is pulled to one side, so that the string makes an angle of 30 with the vertical, and then released from rest. Find the speed of the bob at the lowest point of its swing, the instantaneous angular velocity of the pendulum at that point of its swing, and the tension in the string at that moment. b) 20