Assignment - Periodic Motion Reading: Giancoli, Chapter 5 Holt, Chapter 7 Objectives/HW: The student will be able to: 1 Define and calculate period and frequency. 2 Apply the concepts of position, distance, displacement, speed, velocity, acceleration, and force to circular motion. 3 State and correctly apply the relation between speed, radius, and period for uniform circular motion. 4 State and correctly apply the relation between speed, radius, and centripetal acceleration for form circular motion. 5 Distinguish and explain the concepts of centripetal vs. centrifugal force. 6 State and apply Newton's Law of Universal Gravitation and define and apply the concept of a gravitational field and its relation to the value of g. 7 Combine equations of circular motion and gravitation to solve problems involving orbital motion. HW: 1-13 14 15-25 26-33 Homework Problems 1. (a) Is it possible to go around any type of curve with zero acceleration? Explain. (b) Is it possible to go around any type of curve with a constant acceleration? Explain. (c) If an object goes around a circular curve with a constant speed, will the acceleration be a constant or changing vector quantity? Explain. 2. It takes a 615 kg car 14.3 s to travel at a uniform speed around a circular racetrack of 50.0 m radius. (a) What is the speed of the car? (b) What is the acceleration of the car? (c) What amount of centripetal force must the track exert on the tires to keep the car moving in the circle? 3. An athlete whirls a 7.00 kg hammer tied to the end of a 1.3 m chain in a horizontal circle. The hammer revolves at a frequency of 1.5 Hz. (a) What is the speed of the hammer? (b) What is the centripetal acceleration of the hammer? (c) Ignoring the effect of gravity, what is the tension in the chain,? 4. Sue whirls a yo-yo in a horizontal circle. The yo-yo has a mass of 0.20 kg and is attached to a string 0.80 m long. Ignore the effect of gravity. (a) If the tension in the string is 5.0 N what will be the resulting speed of the yo-yo? (b) Determine the resulting period of the yo-yo s revolution.
5. A coin is placed on a stereo record revolving at 33.3 rpm. (a) In what direction must the coin accelerate to remain on the record? (b) Determine the acceleration rate of the coin when it is placed 5.0, 10, and 15 cm outward from the center of the record. (c) What type of force causes this acceleration? (d) At which of these three radii would the coin be most likely to fly off? Why? 6. A 50.0 gram mass is placed on a level rotating platter 20.0 cm from its center. The coefficient of starting friction is 0.60. Determine the maximum frequency with which the platter can rotate without the mass sliding off its surface. 7. According to a recent edition of the Guinness Book of World Records the highest rotary speed ever attained was 2010 m/s. This speed occurred at one end of a rod 15.3 cm long that rotated about the other end. (a) What was the centripetal acceleration of the free end of the rod? (b) If a 1.00 gram object were attached to the end how much force would be needed to hold it there while rotating at the given rate? (c) What was the frequency of rotation of the rod in revolutions per minute? 8. In a type of amusement park ride sometimes called the Gravitron, the people are spun around inside a large cylinder. If the rate of rotation is great enough the rider may stick to the inside wall of the cylinder with their feet not touching the floor. Suppose the radius of the cylinder is 2.0 m and that it rotates with a frequency of 1.1 Hz. (a) Determine the speed of a rider. (b) Determine the centripetal acceleration of a rider. (c) The force causing this acceleration is what type and exerted by what? (d) Determine the coefficient of friction that is necessary to keep a rider from sliding down the wall. See p. 141, Figure 5-36, for a picture of this type of ride. 9. An early major objection to the idea that the Earth is spinning on its axis was that Earth would turn so fast at the equator that people would be thrown off into space. Show the error in this logic by solving the following problems for a 97.0 kg person standing on the equator where the Earth s radius is 6378 km. Such a person moves in a circular path due to the Earth s rotation about its own axis. (a) Determine the centripetal force necessary to keep this person moving in such a fashion. (b) Determine the pull of gravity acting on the person. (c) Determine the normal force acting on the person. (Note: the person is not at rest unless the Earth s surface is the frame of reference. For this problem, let the Earth s axis be the frame of reference. The person is in motion around the axis even though he is not moving relative to the surface of the Earth.)
10. An automobile wheel and tire is balanced by a 20.0 g piece of lead located on the rim of a 40.6 cm diameter wheel. As the wheel is balanced horizontally on the machine at the tire store it rotates clockwise with a constant frequency of 15.0 Hz. (a) Find the period of the motion. For the instant at which the lead reaches the eastern most part of its circular path find: (b) its velocity, (c) its acceleration, and (d) the net force upon it. (e) What distance does the lead travel in 1.00 minute? 11. A car magazine reports that a 1100 kg car has skid pad results of 0.80 g. This is an indication of the maximum centripetal acceleration possible before the car slides. (a) What is the maximum speed at which this car can go around a level curve with radius 30.0 m? (b) What is the minimum radius of curvature the car can go around at a highway speed of 30.0 m/s? (c) What magnitude of friction is on the car in either of these turns? (d) Show that the coefficient of friction is 0.80 by calculating the friction force divided by the normal force. 12. What is the maximum speed at which a car can travel around a level circular track of radius 80.0 m if the coefficient of static friction between the tire and road is 0.30? (Note: you do not need the car s mass to get the answer. The answer would be the same for any mass car.) 13. In what direction relative to the velocity of an object must a force be applied in order to cause the object to move uniformly in a circle? Explain. 14. While driving your car, if you go around a curve rapidly, you will feel as if you are being pushed away from the center point of the curve. Physicists often refer to this as a fictitious force because there is no actual force pushing you away from the center. Instead there is a real force pulling you toward the center of the curve. Explain using Newton s Laws. 15. The Moon and the Earth are attracted to each other by gravitational force. Does the more massive Earth attract the Moon with a greater force than the Moon attracts the Earth? 16. A spacecraft traveling from Earth to the moon reaches a point where the gravitational attraction of the moon equals that of Earth. Would this location be when the craft is halfway to the moon? Explain. 17. Tom has a mass of 70.0 kg and sally has a mass of 50.0 kg. Tom and Sally are standing 20.0 m apart on the dance floor. Is the handsome young Tom attracted to the pretty young Sally? According to Newton, Yes! Determine the amount of gravitational pull that attracts Tom toward Sally.
18. Two bowling balls each have a mass of 6.8 kg. They are located next to one another with their centers 21.8 cm apart. What amount of gravitational force does one exert on the other? 19. The gravitational force between two electrons 1.00 m apart is 5.42 x 10-71 N. Determine the mass of an electron. 20. Two spherical balls are placed so their centers are 2.6 m apart. The force between the two balls is 2.75 x 10-12 N. What is the mass of each ball if one ball is twice the mass of the other? 21. Use appropriate information from the given Solar System Data Table to determine which object pulls more on the Moon the Earth or the Sun. Which exerts the greater gravitational pull and how many times greater than the other? Assume the distance from the Sun to the Moon is essentially the same as the distance from the Sun to the Earth. 22. The asteroid Ceres has a mass 7.0 x 10 20 kg and a radius of 500 km. (a) What is g on its surface? (b) How much would a 85 kg astronaut weigh on the surface of Ceres? 23. A proposed space station shaped like a huge bicycle wheel with diameter of 625 m is to spin at a rate to produce "artificial gravity" for the occupants in the "rim" of the station. What should be the period of the station's rotation? See p. 138, Figure 5-30 and p. 144, Figure 5-42 for diagrams that illustrate this idea. 24. Suppose the Moon ceased to revolve around Earth. (a) Using information from the Solar System Data Table, find the acceleration each body would have toward the other. (b) As the Moon and Earth got closer together would their rates of acceleration decrease, increase, or remain constant? Explain. (c) The Moon and the Earth do accelerate toward one another at the amounts you found and yet the distance between them does not change! Explain. 25. It is often said the astronauts are "weightless" in orbit. Indeed the amount of gravity will decrease as the astronauts are moved farther from earth. Does it diminish to the extent that it can be dismissed as insignificant? Compare for an astronaut with mass 75.0 kg. (a) Find their weight on earth. (b) Find the magnitude of the force of gravity upon them while aboard the shuttle in orbit 525 km above the earth. (c) The astronauts are not really weightless if by that we mean there is no gravity. So why do they appear to float around inside the space shuttle? Explain using physics concepts. 26. How would you answer the question, What keeps a satellite up? Or in other words if gravity pulls it downward why doesn t it fall to the Earth?
27. Suppose an engineer at NASA is designing a space probe that will orbit the planet Mercury at an altitude of 400 km. In order to calculate the required orbital parameters such as speed, period, etc. what other numerical information will be needed? 28. For a satellite to orbit uniformly about the Earth, what must be true of its speed at orbits of greater and greater radius? 29. A certain satellite has an orbital radius of 4.23 x 10 7 m. (a) Calculate its speed in orbit. (b) Calculate its orbital period expressed in hours. (c) What is this type of orbit called and what makes it uniquely valuable? 30. On July 19, 1969, Apollo 11 s orbit around the Moon was adjusted to an average altitude of 111 km. (a) At that altitude how many minutes did it take to orbit once? (b) At what speed did it orbit the Moon? 31. Determine the mass of the Sun based on its effect on the earth. Earth orbits the Sun at a distance of 150 Gm and completes its circular orbit once every year. (Even though you could just look up the mass of the Sun in a table somewhere it is only by this type of calculation that anyone was ever able to determine this value!) 32. Mimas, a moon of Saturn, has an orbital radius of 187 Mm and an orbital period of 23 hours. Use this information to calculate the mass of Saturn. 33. The space shuttle typically orbits earth (radius 6378 km) in a circular path at an altitude of 525 km. (a) Find the speed of the shuttle. (b) Find the time in minutes to complete one orbit.
Answers Circular Motion and Gravity 1. a. No. It is not possible to curve without accelerating. If there is no acceleration there is no change in velocity and if there is no change in velocity an object will move in a straight line. b. Yes. Only in certain situations would a curve involve constant acceleration. An example is projectile motion. As the projectile follow its parabolic curve it has a constant acceleration equal to g. c. Uniform circular motion involves a changing acceleration vector. The magnitude of the acceleration remains constant in uniform circular motion but the direction of the acceleration changes so that it always point toward the center. 2. a. 22.0 m/s b. 9.65 m/s 2 toward center c. 5940 N 3. a. 12 m/s b. 120 m/s 2 c. 810 N 4. a. 4.5 m/s b. 1.1 s 5. a. The coin must accelerate toward the center of the record. b. 0.61, 1.2, 1.8 m/s 2 c. Friction is the force responsible for the coin s acceleration d. The coin is most likely to fly off at the largest radius because greater acceleration is required and there is only so much friction to cause this acceleration. 6. 0.86 Hz 7. a. 2.64 10 7 m/s 2 b. 26.4 kn toward the center c. 126,000 rpm 8. a. 14 m/s b. 95 m/s 2 c. The force accelerating the rider is the normal force of the wall pushing inward on the person s body. d. 0.10 9. a. 3.27 N b. 951 N c. 948 N, 90 10. a. 66.6 ms b. 19.1 m/s, 270.0 c. 1800 m/s 2, 180.0 d. 36.0 N, 180.0 e. 1.15 km
11. a. 15 m/s b. 110 m c. 8.6 kn d. 0.80 12. 15 m/s 13. In circular motion the velocity is always tangent and the acceleration is always perpendicular to this and pointing toward the center of the circle. Because acceleration is in the direction of force, the force must be perpendicular to the velocity. 14. Your body has inertia and according to Newton s 1 st Law it will continue in a straight line with constant velocity unless there is force to cause otherwise. As your car goes around the curve your body would go straight if not for some force acting on it. It is the seat belt and friction with the seat that applies force to the bottom half of your body and pulls you into the curve as the upper half of your body is merely tending to follow a straight path that would eventually take you out of the curve. There is no real force pushing the top half of your body out from the center; rather, there is a real force pulling the bottom half of your body inward toward the center. Essentially your bottom half is pulled out from under your top half as your top half continues on its way due to its inertia. According to Newton s 2 nd Law there must be a force toward the center to cause your body to accelerate toward the center. 15. The Moon attracts the Earth with a force equal in magnitude to the amount that the Earth attracts the Moon. However, the resulting acceleration of the Moon is greater because of its lesser mass. 16. In order for the two attractive force to be equal the spacecraft would have to be closer to the Moon than to the Earth. This is because the Moon is less massive. When the spacecraft is halfway and equal distance from the two bodies, the Earth will definitely attract it more because of its greater mass. Because of the inverse square aspect of Newton s Law of Universal Gravitation the spacecraft would have to be about ten times closer to the Moon than to the Earth for the forces to be equal assuming the Earth is about 100 times more massive. 17. 0.584 nn 18. 65 nn toward the other ball 19. 9.01 10-31 kg 20. 0.37 kg, 0.75 kg 21. Sun pulls 2.3 times the Earth 22. a. 0.19 m/s 2 b. 16 N 23. 35.5 s 24. a. Earth: 33 mm/s 2 toward Moon Moon: 2.7 mm/s 2 toward Earth b. The accelerations listed above would increase should the Earth and Moon get closer together because the force of gravity increases as the distance decreases.
c. Each of these values are examples of centripetal acceleration. This means that the acceleration describes the rate of change in the body s velocity as it goes around the curve of its orbit. For each object it is only enough acceleration to cause enough turning for the orbit to occur but it is not enough acceleration to bring the objects closer together. Note that the Earth has to move in an orbit about the Moon i.e. it wobbles around in a little circle with the acceleration shown above as the Moon moves in its larger orbit about the Earth. 25. a. 735 N b. 627 N c. The astronaut and the shuttle are both pulled by gravity and both accelerate toward the Earth at the same rate. Because they both accelerate in the same direction at the same rate the astronaut does not accelerate toward the shuttle. For example if the astronaut is in the center of the shuttle cabin he does not get any nearer to the floor or ceiling of the cabin because he is not accelerating in that frame of reference. Therefore he remains floating in the cabin! 26. The satellite s inertia keeps it moving forward in its orbit. Gravity is the force that prevents it from moving in a straight line but it is only enough force to make the satellite s path curve into the shape of a circle. Because the satellite is moving forward so rapidly the force of gravity is not enough to cause it to get any closer to the Earth. 27. The engineer will need to know the mass and radius of Mercury in order to calculate an orbit about it. 28. At a greater radius a satellite must have a lesser speed. The gravitational field gets weaker at higher altitudes if the satellite moves too fast, the gravity will not be strong enough to prevent it from flying out into space. 29. a. 3070 m/s b. 24.0 h c. This is a geosynchronous orbit. The satellite moves in sync with the surface of the Earth and remains above the same point on the Earth. This makes it possible to have continuous communication with the satellite. 30. a. 120 minutes b. 1600 m/s 31. 1.99 x 10 30 kg 32. 5.6 x 10 26 kg 33. a. 7600 m/s (= 17000 mph!) b. 95.1 minutes