Circular Velocity and Centripetal Acceleration

Transcription

1 1. An object is spun around in circular motion such that it completes 100 cycles in 25 s. a. What is the period of its rotation? [0.25 s] b. If the radius is 0.3 m what is the velocity? [7.54 m/s] Circular Velocity and Centripetal Acceleration 2. An object completes 2500 cycles in 25 s. a. What is the period of its rotation? [0.01 s] 3. An object completes 95 cycles in 2.58 minutes. a. What is velocity in m/s if the radius is 0.18 m? [0.694 m/s] 4. An object is spun around in circular motion such that its period is 15 s. a. How much time will be required to complete 58 rotations? [870 s or 14.5 min] b. What is its frequency of rotation? [0.067 Hz]

2 5. An object is spun around in a circle of radius 1.3 m with a period of 4.5 s. a. What is its velocity? [1.81 m/s] b. What is its acceleration? [2.53 m/s 2 ] c. Which direction does the acceleration point? [towards the center] 6. A 5.0 kg object is spun around in a circle of radius 0.85 m with a frequency of 10 Hz. a. What is the period of its rotation? [0.1 s] b. What is its velocity? [53.4 m/s] c. What is its acceleration? [3,355 m/s 2 ] d. What is the net force acting on it? [16,778 n] 7. A 250 kg object is spun around in a circle of radius 20 m with a velocity of 250 m/s. a. What is the period of its rotation? [0.503 s] b. What is its frequency? [1.99 Hz] c. What is its acceleration? [3125 m/s 2 ] d. What is the net force acting on it? [781,250 N]

3 8. A rotating fan completes 1200 revolutions every minute. Consider the tip of a blade, at a radius of 0.15 m. a. Through what distance does the tip move in one revolution? [0.94 m] b. What is the speed of its tip? [18.8 m/s] c. How long does it take for the fan to go around once? [0.05 s] 9. What is the velocity of a speck of clay on the edge of a potter s wheel turning at 45 RPM if the wheel s diameter is 32 cm? [0.754 m/s] 10. A bike tire is spinning in a circle. The radius of the circle is 0.45 m and it takes 3 seconds to go around once. What is the velocity at the edge of the tire? [0.94 m/s]

4 11. A person standing on the edge of a merry-go-round that has a radius of 2.0 m is traveling at 5 m/s. How long does it take for them to complete one lap? 12. A tether ball is spun in circle so that its velocity is 2 m/s. What must be the radius of the circle if it takes 3 second to orbit? [2.5 s] 13. A gymnast is swinging around a bar. He makes 10 revolutions in 4 seconds. a. How long does it take for him to go around once? [0.4 s] b. What is the velocity of his feet (radius = 2.5 m)? [39 m/s] c. What is the velocity of his head (radius = 0.25 m)? [3.9 m/s] [0.95 m]

5 14. A person is on a ride at an amusement park. The ride makes 4 revolutions in 30 seconds. A. What is the period? [7.5 s] B. What is the frequency? [0.13 Hz] C. What is the velocity? [4.2 m/s] D. What is the acceleration? [3.5 m/s 2 ] E. If the students mass is 75 kg. What must be the force? [263 N] 15. A 65 kg person is out riding their bike in a circle with a radius of 22.3 m. In order to stay on the road they need a centripetal force of N. A. What must be their acceleration? [2.87 m/s 2 ] B. How fast must they be traveling? [8.0 m/s] C. What is their period? [17.5 s] D. How many laps will they make in 10 minutes? [34.25 laps] 22.3 m [22.3 m] 16. Another ride at an amusement park makes 15 revolutions in 500 seconds. The radius of the ride is 9 m. What is the velocity of the passengers? [1.7 m/s]

6 FBD s

7 No friction Forces on rider (top and bottom) Forces on pilot (top and bottom)

8 17. A puck on the end of string rotates at 16 RPM. The radius is 0.85 m and the mass of the puck is 1.5 kg. A. What is the velocity of the puck in m/s? [1.42 m/s] B. What will be the tension in the string? [3.58 N] C. How long will it take for the puck the complete 1 revolution? [3.8 s] D. What must be the mass of the block hanging in the center? [0.365 kg] E. What happens to the force if the radius is decreased by ½? (same velocity) [7.1 N] Circular Motion Worksheet #2 18. A coin is placed on a rotating record player. The coin is observed to rotate at 3.5 m/s at a radius of 1.5 m. A. What force provides the centripetal force? [friction] B. What must the coefficient of friction be? [0.83] C. How many revolutions will the coin make in 4 minutes? [89.1 rev]

9 19. The coefficient of friction between a rider and the merry go round is 0.45 and the person is measured to be traveling at 20.0 RPM. At what radius must the person be standing? [1.00 m] 20. A car traveling at 26.4 m/s is rounding a curve with a radius of 75 m. What must the coefficient of friction be between the tires and the road? [0.95] 21. A 75 kg person is on a gravitron ride where the radius is 3.0 m and it spins with a velocity of 6.5 m/s. A. What is the force of gravity on the person? [735 N] B. What is the normal force on the person? Where is it directed? [1056 N, inward] C. What force is canceling out the force of gravity? [friction] D. What must the μ be so the person does not slip down the wall? [0.696]

10 22. A jetliner has a weight of 550,000 N. L is the force of lift on the plane. A. Calculate L. [569,401 N] B. If the plane is traveling at 252 m/s. What is the turn radius? [24,200 m] 23. A car is going over a hill that has a radius of 30 m. The driver s mass is 85 kg. A. What is the normal force exerted by the seat on the driver at the top if v = 15 m/s? [196 N] B. What speed should the car be traveling if the driver is to be weightless at the top? [17.1 m/s] 30 m 24. A rollercoaster is traveling at 20. m/s at the top of a vertical loop. What must be the radius of the loop so the passenger will not fall out at the highest part? [41 m]

11 25. A person is sitting on a swing ride at great adventure. The ride makes an angle of 28º with the vertical. The rope connecting the swing to the disk is 8.0 m long and the radius of the disk is 1.5 m. What is the radius of the rider? What is the ride s velocity in rev/min? [5.26 m] [9.5 rev/min] 28º 26. A gravitron ride has a µ=0.95 and a radius of 5.0 m. What must be the period of the ride so that the floor can drop out and the passengers remain in the air? [4.38 s]

12 27. A 1200 kg car rounds a flat corner with a radius of 45 m and that has a µ equal to a. What force keeps the car on the road? b. How fast can the car go without skidding? [19 m/s] c. What will be the centripetal acceleration of the car? [8.02 m/s 2 ] d. What will be the centripetal force on the car? [9630 N] TOP VIEW 45 m 28. Stewie is on the Taz Twister at great adventure. He is spinning at 9 m/s. The walls of the ride have a µ = What must be the radius of the ride? What force provides the centripetal force? [6.2 m]

13 29. The following is a VERTICAL circle: a. What will be the tension in the cord at the top? [44.1 N]The bottom? [103 N] b. How fast must the object be swung in order for the cord to remain taught when the object is at the top? [greater than 4.4 m/s] 2 m 3 kg 7 m/s 30. What must be the minimum speed of the block at A in for it to make the circle and not fall? 3 m [12.1 m/s]

14 31. How many g s (that is how many times his normal weight) will the pilot pull at the bottom of this loop? If he maintains his speed and completes the loop, how many g s will he feel at the top? [11 g s bottom, 8.9 g s top] 500 m 220 m/s 32. A 0.56 kg plane attached to a 2.00 m string is rotated so that is makes 8.50 revolutions in one second. A. What must be the tension in the string? [3200 N] B. If the string's length is halved what does the tension become (same velocity as A)? [6400 N] C. If the plane rotates at 17 rev/sec (twice as fast) what is the tension (same radius as A)? [13,000 N]

15 33. A 50.0 kg student stands 2.50 meters from the center on a merry-go-round rotating at 3.50 m/s. What must the coefficient of friction be in order for the student to keep from sliding? [0.500] 34. Most humans, with aid of a flight suit, pass out if they are submitted to an acceleration of 9.0 g's. If a jet is traveling at 344 m/s (mach 1), what is the minimum horizontal turn radius a person can survive? [1340 m] 35. In order to simulate gravity in space it has been conceived that a rotating space station would do the job. If the space station has a radius of m, how many revolutions per minute are needed to accelerate at person on the edge at 9.80 m/s 2? [0.863 rev/min]

16 36. A person is sitting on a "swing ride" at Great Adventure. The length of the cable is m and while the ride is running the cable makes and angle of A. What is the velocity of the passenger? [14.8 m/s] B. How many rev's/min will the ride make? [10.9 rev/min] 37. A car is traveling at 29.1 m/s (65 mph) on a flat (unbanked) highway. It enters a turn with a radius of 250. m. What is minimum coefficient of friction needed to keep the car from sliding? TOP VIEW [0.345]

17 38. While in the Gravitron (r = 2.50 m), a person observes the floor to drop out and they see that they are still suspended in the air (pinned against the wall). If μ=0.78 what must be the velocity of the Gravitron in m/s? Rev/min? [5.60 m/s] [21.4 rev/min] 39. A physics student makes a bet that if they can run fast enough they could "run on the walls" in a circular building with a radius of 3.00 m. If the μ=0.81, how fast must they run in order to do so [6.02 m/s] F.Y.I. :[13.5 mph]

18 40. A 84.0 kg person is sitting on a scale in a Ferris Wheel with a radius is 9.00 m and turns once on its axis every 28.0 seconds. A. What does the scale read at the bottom of the loop? [861 N] B. What does the scale read at the top of the loop? [785 N] 41. A stuntman is attempting to make a loop-de-loop with a radius of 2.5 m. A. What must be his a C at the top of the loop to not fall out? [9.8 m/s 2 ] B. How fast (minimum) must he be traveling at the top of the loop to not fall? [4.95 m/s] C. What must be his velocity (minimum) at the bottom of the loop? [11.1 m/s] 2.5 m

19 42. A bullet is fired at 357 m/s into (inelastic collision) a 5.0 kg mass hanging on the end of string that is 1.0 m long. A. What must be the mass of the bullet so the string remains taught at the top? [0.10 kg] B. If an identical bullet is fired into the mass at twice the speed, what will be the tension in the string at the top? [750 N] 1.0 m v = 357 m/s 43. A small plane is banking to make a turn with a radius of 3 m. What must be its velocity? R = 3 m 10º [13 m/s]

20 44. A rider on a Ferris Wheel (R = 15 m) notices that they weight 25% less on the top of the ride than when they are off the ride. a. What is the velocity of the ride? [6.1 m/s] 15 m

21 1. Calculate the force of attraction between a 5.6x10 4 kg mass and a 3.4x10 5 kg mass if they are separated by 2.35 m. Gravity Worksheets 2. The force due to gravity between two objects is N. Determine the force if: A. The distance between them is doubled B. The distance between them is decreased to ⅓ of the original distance? C. What will the force become if one of the masses is quadrupled? [0.230 N] A. [100.0 N] B. [3600. N] C. [1600. N] 3. What force does the moon exert on Mr. McGeechan (m = 86.2 kg) if the mass of the moon is 7.35x10 22 kg and it is orbiting 3.84x10 8 m away? [ N]

22 4. Two masses, one 3 times the other, are placed 3.50 m apart. The experience a force of 1.50 N. What is the mass of each object? 5. Determine the net force on the kg mass. [m 1 = 3.03x10 5 kg, m 2 = 9.09x10 5 kg] [2.32x10-11 N to the left]

23 6. Determine the magnitude and direction of the net force on the 4.00 kg mass in the following: 0.25 m 0.40 m [1.32x10-8 from x-axis] 7. The force of gravity between two bodies is 4.2x10 4 N. The mass of each body is 6.7x10 3 kg. How far apart are the two bodies? [2.67x10-4 m] 8. The force of gravity between two bodies is 6.68x10 3 N. a. What will the force become if the distance between them is tripled? [742 N] b. What will the force become if the mass of one body is halved? [3340 N]

24 9. Calculate the force between two bodies if the mass of one body is 250. kg, and the mass of the other body is 25.0 kg. The distance between the two bodies is 5.50x10 4 m. 10. How far from the surface of the Moon will a spacecraft experience a net gravitational force of zero? R M = 1.74x10 6 m M M = 7.35x10 22 kg [1.38x10-16 N] 3.84x10 8 m [3.66x10 7 m]

25 11. What is the acceleration due to gravity (g) on Jupiter (m J = 1.90x10 27 kg, R J = 6.91x10 7 m)? How much would a 3.00 kg book weigh on Jupiter? 12. What is the acceleration due to gravity (g) on a planet that has a mass of 3M E and radius of ½ R E? [26.5 m/s 2, 79.6 N] 13. The g on a planet is 14.5 m/s 2. The planet has the same mass as the Earth. What is its radius? [118 m/s 2 ] [5.25x10 6 m]

26 14. A satellite is in orbit around the Earth with a radius of 8.98x10 6 m. a. What is the velocity of the satellite? [6664 m/s] b. What is the centripetal acceleration (a C ) of the satellite? [4.96 m/s 2 ] 15. A satellite is in orbit above the Earth at a distance of 4.23x10 7 m from the Earth's center. A. What is the velocity of the satellite? B. What is the satellite s period in days? A. [3070. m/s] B. [1.00 day, geosynchronous orbit!]

27 16. What must the velocity of a satellite be if it is in orbit 110. km above the Earth's surface? 110 km 17. The Earth orbits the Sun once every days (3.16x10 7 seconds) at a radius of 1.50x10 11 m. Use this information to calculate the mass of the Sun. [7840 m/s] [2.00 x10 30 kg]

28 18. An 850 kg satellite is put into orbit at a height of 250 km. Its velocity is 7000 m/s. A. What is the centripetal acceleration of the satellite? [7.39 m/s 2 ] B. Is the satellite in a stable circular orbit? [No] C. Is the satellite moving away from or towards the Earth? [towards] D. Sketch the path that the satellite will take (assuming it won t crash) 19. What must the orbital height (above the surface) of a satellite that is in geosynchronous with a point on the Earth's equator? [3.59x10 7 m]

29 20. A satellite is launched into an orbit that is 250. km above the Earth's surface. A. What velocity must the satellite have? [7756 m/s]] B. What must its period be? [5368 s] C. What is its centripetal acceleration? [9.11 m/s 2 ] KEPLER S LAW PROBLEMS 21. Given that Neptune's orbital radius is 4.49x10 12 m and is nearly circular, how long (in years) will it take for Neptune to complete one orbit? [164 years] A. What is Neptune's velocity in m/s? [5450 m/s] B. What is Neptune's centripetal acceleration? [6.62x10-6 m/s 2 ]

30 22. The orbital period of Mercury is 88.0 days. What is the orbital radius of Mercury in meters? [5.81x10 10 m] 23. Saturn's orbital radius is 9.81 times the orbital radius the Earth. What is Saturn's period in years? [30.7 yr]

31 24. The orbital period of Mars is days. What is the orbital radius of Mars? [2.29x10 11 m] 25. Neptune orbits the Sun with a radius of 4.49x10 12 m. What is Neptune's radius in A.U. (astronomical units)? [30 AU]

32 REVIEW.. Conceptual 1. Since the Earth is constantly attracted to the Sun by the gravitational interaction, why doesn t it fall into the Sun and burn up? 2. As a part of their training before going into orbit, astronauts ride in an airliner that is flown along the parabolic trajectory as a freely falling projectile. Explain why this gives the same apparent weightlessness as being in orbit. 3. A person has a mass of 75.0 kg. What is their weight on Earth? The travel to planet Daygabba where the value of g is 3 times what it is on Earth, what is the person s mass there? What is their weight on Daygabba?

33 4. Why is it impractical to put a satellite in orbit at the Earth s surface? What difficulties are there? 5. The Earth pulls an apple downward. The Earth is pulled upward by the apple (same force). Why does the apple move and not the Earth? 6. What happens to a person s body if they spend long periods in zero-g?

34 7. The diameter of a planet is 5 times that of the Earth s, yet the acceleration due to gravity on both planets is the same. How is this possible? 8. Why can t you launch a satellite straight up and have it orbit? 9. State all three of Kepler s Laws of planetary motion.

35 10. A planet is traveling in an elliptical orbit. When is it traveling the fastest? The slowest? Problems 1. At what altitude above Earth s surface would the gravitational acceleration be 4.9 m/s 2? [2.64x10 6 m]

36 2. The Martian moon Phobos travels in an approximately circular orbit of radius m with a period of7 hr 39 min. Calculate the mass of Mars from this information. [6.47x10 23 kg] 3. The mean distance of Mars from the Sun is 1.52 times that of Earth from the Sun. From Kepler s law, calculate the number of years required for Mars to make one revolution around the Sun; [1.87 yrs]

37 4. 20 kg satellite has a circular orbit with a period of 2.4 hr and a radius of m around a planet of unknown mass. If the magnitude of the gravitational acceleration on the surface of the planet is 8.0 m/s 2, what is the radius of the planet? [5.81x10 6 m] 5. A typical GPS (Global Positioning System) satellite orbits at an altitude of m. Find (a) the orbital speed, and (b) the orbital period of such a satellite [v = 3890 m/s, T = sec]

38 6. An astronaut exploring a distant solar system lands on an unnamed planet with a radius of 3560 km. When the astronaut jumps upward with an initial speed of 3.00 m/s, she rises to a height of m. What is the mass of the planet? [1.50x10 24 kg] 7. The asteroid Icarus, though only a few hundred meters across, orbits the Sun like the planets. Its period is 410 d. What is its average distance from the Sun in meters? In A.U.? [1.62x10 11 m = 1.08 A.U.]

39 8. A satellite of mass 5500 kg orbits the Earth and has a period of 6200s. Find (a) the radius, and (b) kinetic energy of the satellite. [R = 7.3x10 6 m, KE = 1.5x10 11 J] 9. How long would a day be (in minutes) if the Earth were rotating so fast that objects at the equator were apparently weightless? (Hint: Normal force would be zero) [84.5 min]