A. Circular Motion
1. (I) 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. (I) 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. (II) 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. (II) 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 linear speed of the yo-yo?
b) What is the centripetal acceleration of the yo-yo?
c) What is the tension in the string?
5. (II) The same child in question # 4 swings the same yo-yo twice as fast.
a) What is the linear speed of the yo-yo?
b) What is the centripetal acceleration of the yo-yo?
c) What is the tension in the string?
6. (III) A test pilot 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. (II) 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. (II) 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. (II) 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. (II) 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. (II) 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. (II) 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. (II) 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?
1) 0.39 to center / 2) (a) 9.68 m/s2 (b) 5950 N / 3) (a) 630 m/s2 (b) 1260 N4) (a) 1.6 m/s (b) 5.0 m/s2 (c) 1.0 N / 5) (a) 3.2 m/s (b) 20. m/s2 (c) 4.0 N / 6) 161 m
7) Fc = 3.4 N; Fw = 980 N / 8) 0.0026 m/s2 / 9) 3.6 s
10) 0.34 N / 11) Tension doubled / 12) All doubled / 13) v is doubled, a and T are 4 times more
B. Universal Gravitation
1. (I) Two bowling balls, each with a mass of 6.8 kg, are placed next to one another with their centers 21.8 cm apart. What gravitational force do they exert on each other?
2. (II) Two locomotives stand so that their centers are 10 m apart. Each weighs 1.96 × 105 N. What gravitational force exists between them?
3. (I) Using information from the Solar System data table from your reference pack to compute the gravitational force the sun exerts on Jupiter.
4. (I) The mass of the earth is 5.98 × 1024 kg. The moon orbits the earth at a distance (measured from center of the moon to center of the earth) of 3.84 × 108 m, and the gravitational force between them is about 1.98 × 1020 N.
a. What is the approximate mass of the moon?
b. Use Newton’s second law of motion to find the centripetal acceleration given to the moon by the earth’s gravitational force.
c. Therefore, what is the moon’s velocity as it orbits the earth?
5. (I) The mass of an electron is 9.1 × 10−31 kg. The mass of a proton is 1.7 × 10−27 kg. They are about 1.0 × 10−10 m apart in a hydrogen atom. What gravitational force exists between them in the hydrogen atom?
6. (I) Two students are standing 2.0 m apart. One has a mass of 80 kg. The other has a mass of 60 kg. What is the gravitational force between them?
7. (I) Two ships are docked next to each other in Plymouth Harbor with their centers of gravity 40 m apart. One ship weights 9.8 × 107 N, while the other ship weights 1.96 × 108 N. What gravitational force exists between them?
1. 6.49 × 10−8 N / 2. 2.67 × 10−4 N / 3. 4.2 × 1023 N / 4. (a) 7.32 × 1022 kg(b) 0.00270 m/s2
(c) 1,020 m/s
5. 1.0 × 10−47 N / 6. 8.0 × 10−8 N / 7. 8.34 N
C. Mixed Problems
Note: You will need to find some information about the earth, moon, and sun—for example, by looking in the planetary data tables in your reference pack.
1. (I) A child on a carousel is circling at a speed of 1.35 m/s while sitting 3.0 m from the center of the carousel.
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?
c. If the child moves toward the center of the carousel and the angular speed (frequency) of the ride remains the same, will the linear (tangential) speed of the child increase, decrease, or remain the same? Why?
2. (II) 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. (III) 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? If the radius of the turn were increased, would the maximum speed increase, decrease, or remain the same? Why?
4. (III) What coefficient of friction is needed between the tires and the road if a car is to round a level curve of radius 95 m at a speed of 90 km/h?
5. (III) During the Apollo 11 lunar landing mission, the command module orbited the moon at an altitude of 100 km above the surface of the moon. How long did it take the command module to orbit the moon once?
6. (III) 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 down and the riders are pinned to the wall like Spiderman.
a. What coefficient of static friction is necessary to hold the riders in place so that they don’t slip down the wall?
b. If the rate of rotation of the ride were increased, would the minimum coefficient of static friction needed to hold the riders in place increase, decrease, or remain the same? Why?
1) a. 0.6075 m/s2b. 15.19 N
c. decrease / 2) 0.0059 m/s2; gravity / 3) 23.43 m/s; increase
4) 0.6713 / 5) 7070 s (just under 2 hours) / 6) a. 0.1986
b. decrease
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