A. Circular Motion Problems

1) What is the magnitude and direction of the acceleration of a sprinter running at 10.0 m/s when rounding a curve with a radius of 256 m? 2) A 615 kg racing car moving with constant speed completes one lap in 14.3 s around a circular track with a radius of 50.0 m. a) What is the acceleration of the car? b) What force must the track exert on the tires to produce this acceleration? 3) A 2.0 kg mass is attached to a strong that is 1.0 m long and moves in a horizontal circle at a rate of 4.0 revolutions per second. a) What is the centripetal acceleration of the mass? b) What is the tension in the string? 4) A young boy swings a 0.20 kg yo-yo horizontally above his head. The string is 51 cm long and it takes 2.0 s for the yo-yo to make one revolution. a) What is the translational speed of the yo-yo? b) What is the centripetal acceleration of the yo-yo? c) What is the tension in the string? 5) The same child in question # 4 swings the same yo-yo twice as fast. a) What is the translational speed of the yo-yo? b) What is the centripetal acceleration of the yo-yo? c) What is the tension in the string? 6) A test pilot is volunteers to test the limits of a new high-performance fighter plane. The engineers say the jet is capable of flying in a horizontal circle at a speed of 105 m/s. The 80.0 kg pilot does not want his centripetal acceleration to exceed 7 g's (7 times free fall acceleration). What is the minimum radius of the circular path for the plane? 7) An early objection to the idea that the earth is spinning on its axis was that the earth would turn so fast at the equator that people would be thrown into space. Show the error is this logic by calculating the centripetal force needed to hold a 100. kg person in place at the equator. The radius of the earth is about 6400 km. Compare this force with the force of gravity (weight) of the 100. kg person. 8) The radius of the moon’s orbit is about 3.6 x 108 m. The moon’s period of revolution is 27.3 days. Calculate the centripetal acceleration of the moon around the earth. 9) A dog sits 0.50 m from the center of a merry-go-round. If the dog’s centripetal acceleration is 1.5 m/s2, how long does it take the dog to go around once? 10) A 13 g stopper is attached to a 93 cm string. The stopper is swung in a horizontal circle, making one revolution in 1.18 s. Find the tension in the string on the stopper. 11) If the mass of the stopper in problem # 11 is doubled but all other quantities remain the same. What would be the effect on the velocity, acceleration, and tension? 12) If the radius of the circle for the stopper in problem #11 is doubled but all other quantities remain the same, what would be the effect on the velocity, acceleration, and tension? 13) If the period of revolution in problem #11 is half as large but all other quantities remain the same, what would be the effect on the velocity, acceleration, and the force?

13.1) 0.39 to center 13.2) (a) 9.68 m/s2 (b) 5950 N 13.3) (a) 630 m/s2 (b) 1260 N 13.4) (a) 1.6 m/s (b) 5.0 m/s2 13.5) (a) 3.2 m/s (b) 20. m/s2 (c) 4.0 N 13.6) 161 (c) 1.0 N m 2 13.7) Fc = 3.4 N; Fw = 980 N 13.8) 0.0026 m/s 13.9) 3.6 s 13.10) 0.34 13.11) Tension 13.12) All doubled 13.13) v is doubled, a and T are 4 N doubled times more

B. Complex Circular Motion Problems. Use the table below to help with questions in this section. Solar System Data Name Average Radius (m) Mass (kg) Mean Distance from the Sun (m) Center to Center Sun 6.96 x 108 1.99 x 1030 - Mercury 2.44 x 106 3.30 x 1023 5.79 x 1010 Venus 6.05 x 106 4.87 x 1024 1.08 x 1011 Earth 6.38 x 106 5.97 x 1024 1.50 x 1011 Mars 3.40 x 106 6.42 x 1023 2.28 x 1011 Jupiter 7.15 x 107 1.90 x 1027 7.78 x 1011 Saturn 6.03 x 107 5.69 x 1026 1.43 x 1012 Uranus 2.56 x 107 8.66 x 1025 2.87 x 1012 Neptune 2.48 x 107 1.03 x 1026 4.50 x 1012 Pluto 1.15 x 106 1.50 x 1022 5.91 x 1012 Earth’s Moon 1.74 x 106 7.35 x 1022 -

1) A child on a merry-go-round is moving at a speed of 1.35 m/s when 12.0 m from the center of the merry-go-round. a. What is the centripetal acceleration of the child? b. If the mass of the child is 25 kg, what is the net horizontal force exerted on the child? 2) What is the centripetal acceleration given to the Earth by the Sun? What is the real force that causes the centripetal acceleration of the Earth? 3) What is the maximum speed with which a 1000 kg car can round a flat turn with an 80 m radius if the coefficient of friction between the tires and the road is 0.70? 4) What is the coefficient of friction between the tires and the road if a car is to round a level curve of radius 95 m at a speed of 90.0 km/h? 5) During the Apollo 11 lunar landing mission, the command module orbited the moon at an altitude of 100,000 m above the surface of the moon. How long did it take the command module to orbit the moon once? 6) In an “Open Bucket” ride at a carnival, people are rotated in a vertical cylinder “room”. The room has a radius of 5.0 m and is rotating at a frequency of 0.50 rev/s when the floor drops out. What is the coefficient of static friction that holds the rider in place so that they don’t slip down the wall? 7) A car is traveling around a banked curve or radius r at a constant speed v. Determine a formula to determine the maximum angle at which the curve can be banked so that no friction is required. 8) A car is exiting the expressway on an off-ramp curve with a radius of 50 m. What is the maximum ‘design speed’ for the ramp if it is banked at an angle of 22⁰? 9) A highway curve has a radius of 31 m and is banked at 16⁰ with the horizontal. Consider a car of mass 1800 kg whose tires have a coefficient of static friction of 0.61 against the pavement. How fast can the car travel without skidding as it travels around the curve?

1) a. 0.152 m/s2 2) 5.97 x 10-3 m/s2 3) 23 m/s b. 3.8 N 4) 0.67 5) 7059 s approx. 2 6) 0.2 hours 7) 8) 14 m/s 9) 18.6 m/s