DYNAMICS OF UNIFORM CIRCULAR MOTION


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1 chapter DYNAMICS OF UNIFORM CIRCULAR MOTION Section 5.1 Uniform Circular Motion Section 5.2 Centripetal Acceleration 1. A ball moves with a constant speed of 4 m/s around a circle of radius 0.25 m. What is the period of the motion? (a) 0.1 s (c) 0.7 s (e) 2 s (b) 0.4 s (d) 1 s 2. The second hand on a watch has a length of 4.50 mm and makes one revolution in s. What is the speed of the end of the second hand as it moves in uniform circular motion? (a) m/s (c) m/s (e) m/s (b) m/s (d) m/s 3. Approximately one billion years ago, the Moon orbited the Earth much closer than it does today. The radius of the orbit was only km. Today, the radius is km. The orbital period was only s. The present period is s. Assume that the orbit of the Moon is circular. Calculate the ratio of the speed of the Moon in its ancient orbit to the speed that it has today. (a) 15.8 (c) 10.2 (e) 6.39 (b) 12.8 (d) A racecar is traveling at constant speed around a circular track. What happens to the centripetal acceleration of the car if the speed is doubled? (a) The centripetal acceleration remains the same. (b) The centripetal acceleration increases by a factor of 2. (c) The centripetal acceleration increases by a factor of 4. (d) The centripetal acceleration is decreased by a factor of onehalf. (e) The centripetal acceleration is decreased by a factor of onefourth 5. A ball is whirled on the end of a string in a horizontal circle of radius R at constant speed v. Complete the following statement: The centripetal acceleration of the ball can be increased by a factor of 4 by (a) keeping the speed fixed and increasing the radius by a factor of 4. (b) keeping the radius fixed and increasing the speed by a factor of 4. (c) keeping the radius fixed and increasing the period by a factor of 4. (d) keeping the radius fixed and decreasing the period by a factor of 4. (e) keeping the speed fixed and decreasing the radius by a factor of A rock is whirled on the end of a string in a horizontal circle of radius R with a constant period T. If the radius of the circle is reduced to R/2, while the period remains T, what happens to the centripetal acceleration of the rock? (a) The centripetal acceleration remains the same. (b) The centripetal acceleration increases by a factor of 2. (c) The centripetal acceleration increases by a factor of 4. (d) The centripetal acceleration decreases by a factor of 2. (e) The centripetal acceleration decreases by a factor of 4.
2 Physics, 7e TEST BANK A car traveling at 20 m/s rounds a curve so that its centripetal acceleration is 5 m/s 2. What is the radius of the curve? (a) 4 m (c) 80 m (e) 640 m (b) 8 m (d) 160 m 8. A satellite is placed in a circular orbit to observe the surface of Mars from an altitude of 144 km. The equatorial radius of Mars is 3397 km. If the speed of the satellite is 3480 m/s, what is the magnitude of the centripetal acceleration of the satellite? (a) 2.17 m/s 2 (c) 2.99 m/s 2 (e) 4.05 m/s 2 (b) 2.60 m/s 2 (d) 3.42 m/s 2 Questions 9 and 10 pertain to the following situation: One of the world s largest Ferris wheels, the Cosmo Clock 21 with a radius of 50.0 m is located in Yokohama City, Japan. Each of the sixty gondolas on the wheel takes 1.00 minute to complete one revolution when it is running at full speed. Note: Ignore gravitational effects. 9. What is the uniform speed of a gondola when the Ferris wheel is running at full speed? (a) 314 m/s (c) 10.5 m/s (e) 5.24 m/s (b) 1.67 m/s (d) 18.6 m/s 10. What is the centripetal acceleration of the gondola when the Ferris wheel is running at full speed? (a) m/s 2 (c) 2.21 m/s 2 (e) 6.28 m/s 2 (b) 6.91 m/s 2 (d) m/s 2 Section 5.3 Centripetal Force 11. A boy is whirling a stone around his head by means of a string. The string makes one complete revolution every second; and the tension in the string is F T. The boy then speeds up the stone, keeping the radius of the circle unchanged, so that the string makes two complete revolutions every second. What happens to the tension in the sting? (a) The tension is unchanged. (b) The tension reduces to half of its original value. (c) The tension increases to twice its original value. (d) The tension increases to four times its original value. (e) The tension reduces to onefourth of its original value. 12. A 0.25kg ball attached to a string is rotating in a horizontal circle of radius 0.5 m. If the ball revolves twice every second, what is the tension in the string? (a) 2 N (c) 7 N (e) 20 N (b) 5 N (d) 10 N 13. A certain string just breaks when it is under 25 N of tension. A boy uses this string to whirl a 2kg stone in a horizontal circle of radius 3 m. The boy continuously increases the speed of the stone. At approximately what speed will the string break? (a) 6 m/s (c) 12 m/s (e) 18 m/s (b) 9 m/s (d) 15 m/s
3 56 Chapter 5 Dynamics of Uniform Circular Motion 14. Sara puts a box into the trunk of her car. Later, she drives around an unbanked curve that has a radius of 48 m. The speed of the car on the curve is 16 m/s, but the box remains stationary relative to the floor of the trunk. Determine the minimum coefficient of static friction for the box on the floor of the trunk. (a) 0.42 (d) 0.33 (b) 0.54 (e) This cannot be determined without knowing the mass of (c) 0.17 the box. 15. In an amusement park ride, a small child stands against the wall of a cylindrical room that is then made to rotate. The floor drops downward and the child remains pinned against the wall. If the radius of the device is 2.15 m and the relevant coefficient of friction between the child and the wall is 0.400, with what minimum speed is the child moving if he is to remain pinned against the wall? (a) 7.26 m/s (c) 12.1 m/s (e) 9.80 m/s (b) 3.93 m/s (d) 5.18 m/s Section 5.4 Banked Curves 16. Which force is responsible for holding a car in a frictionless banked curve? (a) the reaction force to the car's weight (b) the vertical component of the car's weight (c) the vertical component of the normal force (d) the horizontal component of the car's weight (e) the horizontal component of the normal force 17. Which force is responsible for holding a car in an unbanked curve? (a) the car's weight (b) the force of friction (c) the reaction force to the car's weight (d) the vertical component of the normal force (e) the horizontal component of the normal force 18. Complete the following statement: The maximum speed at which a car can safely negotiate an unbanked curve depends on all of the following factors except (a) the diameter of the curve. (b) the acceleration due to gravity. (c) the coefficient of static friction between the road and the tires. (d) the coefficient of kinetic friction between the road and the tires. (e) the ratio of the static frictional force between the road and the tires and the normal force exerted on the car. 19. Complete the following statement: The maximum speed at which a car can safely negotiate a frictionless banked curve depends on all of the following except (a) the mass of the car. (b) the angle of banking. (c) the diameter of the curve. (d) the radius of the curve. (e) the acceleration due to gravity. 20. Determine the minimum angle at which a roadbed should be banked so that a car traveling at 20.0 m/s can safely negotiate the curve if the radius of the curve is m. (a) (c) 11.5 (e) 78.2 (b) (d) 19.6
4 Physics, 7e TEST BANK A car enters a horizontal, curved roadbed of radius 50 m. The coefficient of static friction between the tires and the roadbed is What is the maximum speed with which the car can safely negotiate the unbanked curve? (a) 5 m/s (c) 20 m/s (e) 100 m/s (b) 10 m/s (d) 40 m/s 22. An indoor track is to be designed such that each end is a banked semicircle with a radius of 24 m. What should the banking angle be for a person running at speed v = 6.0 m/s? (a) 8.7 (c) 14 (e) 45 (b) 11 (d) 22 Questions 23 through 25 pertain to the statement below: A 1000kg Jeep travels along a straight 500m portion of highway (from A to B) at a constant speed of 10 m/s. At B, the Jeep encounters an unbanked curve of radius 50 m. The Jeep follows the road from B to C traveling at a constant speed of 10 m/s while the direction of the Jeep changes from east to south. A W 10 m/s N E S B 50 m C 23. What is the magnitude of the acceleration of the Jeep as it travels from A to B? (a) 2 m/s 2 (c) 10 m/s 2 (e) zero m/s 2 (b) 5 m/s 2 (d) 20 m/s What is the magnitude of the acceleration of the Jeep as it travels from B to C? (a) 2 m/s 2 (c) 10 m/s 2 (e) zero m/s 2 (b) 5 m/s 2 (d) 20 m/s What is the magnitude of the frictional force between the tires and the road as the Jeep negotiates the curve from B to C? (a) N (c) 5000 N (e) 1000 N (b) N (d) 2000 N Section 5.5 Satellites in Circular Orbit Section 5.6 Apparent Weightlessness and Artificial Gravity 26. The earth exerts the necessary centripetal force on an orbiting satellite to keep it moving in a circle at constant speed. Which one of the following statements best explains why the speed of the satellite does not change although there is a net force exerted on it? (a) The satellite is in equilibrium. (b) The acceleration of the satellite is zero m/s 2. (c) The centripetal force has magnitude mv 2 /r. (d) The centripetal force is canceled by the reaction force. (e) The centripetal force is always perpendicular to the velocity.
5 58 Chapter 5 Dynamics of Uniform Circular Motion 27. Callisto and Io are two of Jupiter's satellites. The distance from Callisto to the center of Jupiter is approximately 4.5 times farther than the distance from Io to the center of Jupiter. How does Callisto's orbital period, T C, compare to that of Io, T I? (a) T C = 4.5 T I (c) T C = 9.5 T I (e) T C = 2.7 T I (b) T C = 21 T I (d) T C = 0.2 T I 28. Consider a hypothetical planet in our solar system whose average distance from the Sun is about four times that of Earth. Determine the orbital period for this hypothetical planet. (a) 0.25 year (c) 4 years (e) 16 years (b) 2.5 years (d) 8 years 29. Consider a satellite in a circular orbit around the Earth. If it were at an altitude equal to twice the radius of the Earth, 2R E, how would its speed v relate to the Earth's radius R E, and the magnitude g of the acceleration due to gravity on the Earth's surface? (a) 2 gre 2 gre v = (c) v = 9 3 (e) (b) 2 2 gre v = 2gRE (d) v = 4 2 gre v = A satellite is placed in equatorial orbit above Mars, which has a radius of 3397 km and a mass M M = kg. The mission of the satellite is to observe the Martian climate from an altitude of 488 km. What is the orbital period of the satellite? (a) s (c) s (e) s (b) s (d) s 31. A satellite in orbit around the earth has a period of one hour. An identical satellite is placed in an orbit having a radius that is nine times larger than that of the first satellite. What is the period of the second satellite? (a) 0.04 h (c) 4 h (e) 27 h (b) 3 h (d) 9 h 32. The orbital radius of Saturn about the Sun is about 10 times that of Earth. Complete the following statement: The period of Saturn is about (a) 10 yr. (c) 40 yr. (e) 160 yr. (b) 30 yr. (d) 90 yr. 33. An artificial satellite in a circular orbit around the Sun has a period of 8 years. Determine the ratio of the satellite's orbital radius about the Sun to the earth's orbital radius about the Sun. Assume that the earth's orbit about the Sun is circular. (a) 1 (c) 4 (e) 23 (b) 2 (d) The mass and radius of the moon are kg and m, respectively. What is the weight of a 1.0kg object on the surface of the moon? (a) 1.0 N (c) 3.7 N (e) 9.8 N (b) 1.7 N (d) 8.8 N 35. An object weighs 10 N on the earth's surface. What is the weight of the object on a planet that has one tenth the earth's mass and one half the earth's radius? (a) 4 N (c) 1 N (e) 20 N (b) 2 N (d) 10 N
6 Physics, 7e TEST BANK 59 Questions 36 through 38 pertain to the situation described below: A 2400kg satellite is in a circular orbit around a planet. The satellite travels with a constant speed of m/s. The radius of the circular orbit is m. satellite planet 36. At the instant shown in the figure, which arrow indicates the direction of the net force on the satellite? (a) (b) (c) (d) (e) 37. What is the acceleration of the satellite? (a) 2.5 m/s 2 (c) 9.8 m/s 2 (e) zero m/s 2 (b) 21 m/s 2 (d) 5.0 m/s Determine the magnitude of the gravitational force exerted on the satellite by the planet. (a) N (d) N (b) N (e) This cannot be determined since the mass (c) N and radius of the planet are not specified. 39. What is the acceleration due to gravity at an altitude of m above the earth's surface? Note: the radius of the earth is m. (a) 3.99 m/s 2 (c) 5.00 m/s 2 (e) 7.32 m/s 2 (b) 9.80 m/s 2 (d) 6.77 m/s The radius of the earth is m and its mass is kg. What is the acceleration due to gravity at a height of m above the earth's surface? (a) 1.08 m/s 2 (c) 9.80 m/s 2 (e) m/s 2 (b) 2.15 m/s 2 (d) m/s A spaceship is in orbit around the earth at an altitude of miles. Which one of the following statements best explains why the astronauts experience weightlessness? (a) The centripetal force of the earth on the astronaut in orbit is zero newtons. (b) The pull of the earth on the spaceship is canceled by the pull of the other planets. (c) The spaceship is in free fall and its floor cannot press upwards on the astronauts. (d) The force of gravity decreases as the inverse square of the distance from the earth's center. (e) The force of the earth on the spaceship and the force of the spaceship on the earth cancel because they are equal in magnitude but opposite in direction. 42. A space station is designed in the shape of a large, hollow donut that is uniformly rotating. The outer radius of the station is 350 m. With what period must the station rotate so that a person sitting on the outer wall experiences artificial gravity, i.e. an acceleration of 9.8 m/s 2? (a) 230 s (c) 110 s (e) 38 s (b) 170 s (d) 76 s
7 60 Chapter 5 Dynamics of Uniform Circular Motion 43. The radius of the earth is 6400 km. An incoming meteorite approaches the earth along the trajectory shown. The point C in the figure is 6400 km above the earth s surface. The point A is located at the earth s center. At point C, what acceleration would the meteorite experience due to the earth s gravity? A C 30 B (a) 9.8 m/s 2 toward A (c) 2.5 m/s 2 toward B (e) 5.0 m/s 2 toward A (b) 2.5 m/s 2 toward A (d) 5.0 m/s 2 toward B *Section 5.7 Vertical Circular Motion 44. A plane is traveling at 200 m/s following the arc of a vertical circle of radius R. At the top of its path, the passengers experience weightlessness. To one significant figure, what is the value of R? (a) 200 m (d) 4000 m (b) 1000 m (e) m (c) 2000 m R 200 m/s 45. A 25kg box is sliding down an icecovered hill. When it reaches point A, the box is moving at 11 m/s. Point A is at the bottom of a circular arc that has a radius R = 7.5 m. What is the magnitude of the normal force on the box at Point A? (a) 250 N (d) 650 N (b) 280 N (e) 900 N (c) 400 N R A 46. A 0.75kg ball is attached to a 1.0m rope and whirled in a vertical circle. The rope will break when the tension exceeds 450 N. What is the maximum speed the ball can have at the bottom of the circle without breaking the rope? (a) 24 m/s (c) 32 m/s (e) 38 m/s (b) 12 m/s (d) 16 m/s Questions 47 and 48 pertain to the situation described below: A small car of mass M travels along a straight, horizontal track. R As suggested in the figure, the track then bends into a vertical circle of radius R. M 47. What is the minimum acceleration that the car must have at the top of the track if it is to remain in contact with the track? (a) 4.9 m/s 2, downward (c) 9.8 m/s 2, downward (e) 19.6 m/s 2, upward (b) 4.9 m/s 2, upward (d) 9.8 m/s 2, upward
8 Physics, 7e TEST BANK Which one of the following expressions determines the minimum speed that the car must have at the top of the track if it is to remain in contact with the track? (a) v = MgR (c) v 2 = 2gR (e) v = gr (b) v = 2gR (d) v 2 = gr Additional Problems Questions 49 through 52 pertain to the situation described below: A 1500kg SUV travels at a constant speed of 22 m/s around a circular track that has a radius of 85 m. 49. Which statement is true concerning this SUV? (a) The velocity of the SUV is changing. (b) The SUV is characterized by constant velocity. (c) The SUV is characterized by constant acceleration. (d) The SUV has a velocity vector that points along the radius of the circle. (e) The SUV has an acceleration vector that is tangent to the circle at all times. 50. What is the magnitude of the acceleration of the SUV? (a) 5.7 m/s 2 (c) 9.8 m/s 2 (e) zero m/s 2 (b) 0.26 m/s 2 (d) 1.2 m/s What is the average velocity of the SUV during one revolution? (a) 8.0 m/s (c) 26 m/s (e) zero m/s (b) 12 m/s (d) 44 m/s 52. Determine the magnitude of the net force that acts on the SUV. (a) 390 N (c) N (e) zero newtons (b) 1800 N (d) N 53. Jupiter has a mass that is roughly 320 times that of the Earth and a radius equal to 11 times that of the Earth. What is the acceleration due to gravity on the surface of Jupiter? (a) 2.7 m/s 2 (c) 26 m/s 2 (e) 260 m/s 2 (b) 9.8 m/s 2 (d) 87 m/s 2 Questions 54 through 56 pertain to the situation described below: A rocket orbits a planet in a circular orbit at a constant speed as shown in the drawing. Note these arrows: [1] [2] [3] [4] [5] 54. At the instant shown in the drawing, which arrow indicates the direction of the acceleration of the rocket? (a) 1 (c) 3 (e) 5 (b) 2 (d) 4
9 62 Chapter 5 Dynamics of Uniform Circular Motion 55. At the instant shown in the drawing, which arrow shows the direction of the reaction force exerted on the planet by the rocket? (a) 1 (c) 3 (e) 5 (b) 2 (d) Suppose that the radius of the circular path is r when the speed of the rocket is v and the acceleration of the rocket has magnitude a. If the radius and speed are increased to 2r and 2v respectively, what is the magnitude of the rocket's subsequent acceleration? a (a) (c) a (e) 8a 2 (b) 2a (d) 4a 57. The record for the highest speed achieved in a laboratory for a uniformly rotating object was m/s for a 0.15m long carbon rod. What was the period of rotation of the rod? (a) s (c) s (e) s (b) s (d) s Questions 58 and 59 pertain to the following situation. An airplane flying at 115 m/s due east makes a gradual turn following a circular path to fly south. The turn takes 15 seconds to complete. 58. What is the radius of the curve that the plane follows in making the turn? (a) 280 m (c) 830 m (e) 1600 m (b) 350 m (d) 1100 m 59. What is the magnitude of the centripetal acceleration during the turn? (a) zero m/s 2 (c) 8.1 m/s 2 (e) 12 m/s 2 (b) 6.9 m/s 2 (d) 9.8 m/s 2
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