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1 Unit 6. Circular Motion and Gravitation Name: I have not failed. I've just found 10,000 ways that won't work.-- Thomas Edison Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Essential Knowledge 1.C.2: Gravitational mass is the property of an object or a system that determines the strength of the gravitational interaction with other objects, systems, or gravitational fields. a. The gravitational mass of an object determines the amount of force exerted on the object by a gravitational field. b. Near the Earth s surface, all objects fall (in a vacuum) with the same acceleration, regardless of their inertial mass. Big Idea 2: Fields existing in space can be used to explain interactions. Essential Knowledge 2.B.2: The gravitational Learning Objective (2.B.2.1): field caused by a spherically symmetric object The student is able to apply g=gm/r 2 to calculate the with mass is radial and, outside the object gravitational field due to an object with mass M, where the field varies as the inverse square of the radial is a vector directed toward the center of the object of mass M. distance from the center of that object. Learning Objective (2.B.2.2): a. The gravitational field caused by a The student is able to approximate a numerical value of the spherically symmetric object is a vector whose gravitational field (g) near the surface of an object from its magnitude outside the object is equal to G M/r 2. radius and mass relative to those of the Earth or other reference b. Only spherically symmetric objects will be objects. considered as sources of the gravitational field. Big Idea 3: The interactions of an object with other objects can be described by forces. Essential Knowledge 3.A.2: Forces are described by vectors. a. Forces are detected by their influence on the motion of an object. b. Forces have magnitude and direction situation. Essential Knowledge 3.B.1: If an object of interest interacts with several other objects, the net force is the vector sum of the individual forces. Essential Knowledge 3.C.1: Gravitational force describes the interaction of one object that has mass with another object that has mass. a. The gravitational force is always attractive. b. The magnitude of force between two spherically symmetric objects of mass m1 and m2 is Gm 1 m 2 /r2 where r is the center-to-center distance between the objects. c. In a narrow range of heights above the Earth s surface, the local gravitational field, g, is approximately constant. Learning Objective (3.A.2.1): The student is able to represent forces in diagrams or mathematically using appropriately labeled vectors with magnitude, direction, and units during the analysis of a Learning Objective (3.B.1.1): The student is able to predict the motion of an object subject to forces exerted by several objects using an application of Newton s second law in a variety of physical situations with acceleration in one dimension (including objects in circular motion) Big Idea 4: Interactions between systems can result in changes in those systems. Essential Knowledge 4.A.2: The acceleration is equal to the rate of change of velocity with time, and velocity is equal to the rate of change of position with time. a. The acceleration of the center of mass of a system is directly proportional to the net force exerted on it by all objects interacting with the system and inversely proportional to the mass of the system. Learning Objective (3.C.1.1): The student is able to use Newton s law of gravitation to calculate the gravitational force the two objects exert on each other and use that force in contexts other than orbital motion. Learning Objective (3.C.1.2): The student is able to use Newton s law of gravitation to calculate the gravitational force between two objects and use that force in contexts involving orbital motion (for circular orbital motion only in Physics 1). Students can relate the radius of a circle and the speed or rate of revolution of a particle to the magnitude of the centripetal acceleration. Students can describe and determine the magnitude and direction of the particles velocity and acceleration vectors at any instant during rotation. Proficient 1

2 Circular Motion and Gravitation Reading Assignment Read Chapter 6. As you read answer all Stop to Think questions (Check your answers on page 188) and work through all example problems. Below is a list of what you need to take away from your reading. 1. Define/Know a. Period of rotation b. The equation for speed (v) of an object in uniform circular motion in terms of? c. The equation that relates velocity to acceleration in a circle. d. The equation used to calculate force in circular motion. e. The direction of the velocity, acceleration, and force vectors in uniform circular motion. f. Critical speed g. Orbit h. Newton s Law of gravity i. The gravitational constant j. The equation for calculating the speed of a satellite in circular orbit. (and how to derive the formula using circular motion and newton s law of gravity equations) 2. Explain a. Why points on the outside of the disk travel faster than points near the inside of the disk if they are traveling through the same angle (see example 6.2)? b. Why an object traveling in uniform circular motion has changing velocity c. When a car turns a corner, what force keeps it on the road and why is it harder to make a turn if the road is wet or icy? d. Why does banking a curve (raising the outer edge) increase the speed at which a car can travel through a turn? e. How the magnitude of forces change in a vertical circle (if you are whirling an object on a rope in a vertical circle when it is most likely to break). f. Why an orbiting object is really in free fall. g. Why objects feel weightless in orbit. h. How the force of gravity varies with mass and distance. i. Why a geostationary satellite has to be so far above the earth. j. Why, if the Milky Way Galaxy is held together by the attractive force of gravity, it doesn t collapse on itself. 3. Be able to: a. Calculate velocity, acceleration, and forces in uniform circular motion. b. Draw free body diagram for an object whirled on the end of a string in a horizontal circle and a vertical string. c. Draw free body diagram for a passenger in a roller coaster at the top and bottom of a loop-theloop. d. Explain apparent weight change as an object travels in a vertical circle. e. Calculate critical speed for an object in a circle. f. Calculate the minimum speed an object must have in order to stay in orbit (v orbit ) g. Calculate the acceleration due to gravity on other planets. h. Calculate the speed of a satellite in circular orbit around a planet of mass M. 2

3 Horizontal Circles: 1. A DVD rotates at 988 rpm. How much time is required for one revolution of the DVD? 2. A CD rotates at 540 rpm. How much time is required for one revolution of the CD? In the above picture the distance from the center of the disk to point 1 and 2 is 3 cm and to point 3 is 5 cm. If the disk makes one revolution is 0.5 second, calculate the tangential velocity at each point

4 7. A ball with a mass of 150-g is revolving uniformly in a horizontal circle with a radius of m. The ball makes exactly 2 revolutions in a second. a) Find the tangential velocity of the ball. b) Find the centripetal acceleration of the ball. c) Find the magnitude of the force causing this acceleration. 8. Let us assume that the orbit of the moon around the earth is circular. If the radius of this orbit is 384,000- km and has a period of 27.3 days; a) Find the tangential velocity of the moon (in m/s). b) Find the centripetal acceleration of the moon towards earth. c) If the mass of the moon is kilograms, find the magnitude of the gravitational force keeping the moon going in the circle. 9. A ball with a mass is kg, is moving in a circular path. The radius is 135 meters and the linear speed is 4.50 m/sec. What is the magnitude of the centripetal acceleration of this ball? 4

5 10. A coin of mass m is placed at a distance, r, from the center of a turntable. The coefficient of static friction between the coin and the turntable is s. Starting from rest, the turntable is gradually rotated faster and faster. a. Draw the turntable both as seen from above and as an edge view with the coin on the left side coming toward you. Label the radius, r. b. On the edge view above, draw a free-body diagram of all of the forces acting on the coin (there should be three forces) c. Why does the static frictional force point in the direction you drew it above? d. Under what condition does the coin just start to slip? Write this condition as a mathematical statement. e. Write the Newton s second law equation for the x- and y- components of all forces acting on the coin. One sum should equal 0 and the other mv 2 /r. f. The two equations of part e are valid for any speed up to the point of the slipping. If you combine these with your statement in part d, you can solve for the speed v max at which the coin slips. Do so. g. What is the frequency at which the coin slips? Use the relationship between v and f to find this. 5

6 11. A 1200 kg car is moving on a flat circular track that has a radius of 0.50 km. The car is traveling at 88 km/h. a. What is the name of the force that keeps the car from slipping? b. Calculate the magnitude of this force? c. If this speed is the maximum the car can travel and still stay in the circle, what is the coefficient of static friction for the tires and the road surface? 12. A mouse is sitting on a record player. The mouse is 8.0 cm from the center of the record. Suddenly the record player begins to play at 45 rpm (revolution per minute). The mouse begins to run in place. (a) How fast must it run? (b) The mouse sits down and, amazingly, is able to sit on the record without slipping off. If the mouse is just on the verge of sliding, what is the coefficient of static friction between it and the record? 13. There is an amusement park ride called the ROTOR where you enter a cylindrical room. The room begins to spin very fast until at some point the floor beneath you falls out. Suppose that this room has a radius of 4.20 meters and the room rotates such that you make one complete revolution in 3.65 seconds. a. What will be your linear speed as the room spins? b. What is the magnitude of your centripetal acceleration? 6

7 Vertical Circles: A stunt plane does a series of vertical loop-the-loops. At what point in the circle does the pilot feel the heaviest? Explain. Include a free-body diagram with your explanation. 7

8 You can swing a ball on a string in a vertical circle if you swing it fast enough. a. Draw a free-body diagram for the ball at the top of the circle. b. How does that free-body diagram change if the ball is at the minimum speed in order that it just makes it through the top of the circle? (what is the tension in the string when it is at critical velocity?) 8

9 21. A 0.5-kilogram object rotates at constant speed in a vertical circle at the end of a string of length 2 meters as shown above. As the object passes through point P at the top of the circular path, the tension in the string is 20 N. a. On the diagram of the object above, draw and clearly label all significant forces on the object when it is at point P. b. Calculate the speed of the object at point P. c. Calculate the tension in the string as the object passes through point Q. 9

10 22. A small block of mass 0.15 kg is placed at point A at a height 2.0 m above the bottom of a track, as shown in the figure above, and is released from rest. It slides with negligible friction down the track, around the inside of the loop of radius 0.60 m, and leaves the track at point C at a height 0.50 m above the bottom of the track. a. On the figure below, draw and label the forces (not components) that act on the block when it is at the top of the loop at point B. b. Calculate the minimum speed the block can have at point B without losing contact with the track? (2.4 m/s) c. Calculate the minimum height h min above the bottom of the track at which the block can be released and still go around the loop without losing contact with the track. (1.5 m) 10

11 23. A designer is working on a new roller coaster, and she begins by making a scale model. On this model, a car of total mass 0.50 kg moves with negligible friction along the track shown in the figure above. The car is given an initial speed v 0 = 1.5 m/s at the top of the first hill of height 2.0 m. Point A is located at a height of 1.9 m at the top of the second hill, the upper part of which is a circular arc of radius 0.95 m. a. Calculate the speed of the car at point A. (2 m/s) b. On the figure of the car below, draw and label vectors to represent the forces on the car at point A. c. Calculate the magnitude of the force of the track on the car at point A. (2.7 N) d. In order to stop the car at point A, some friction must be introduced. Calculate the work that must be done by the friction force in order to stop the car at point A. (-1.1 J) e. Explain how to modify the track design to cause the car to lose contact with the track at point A before descending down the track. Justify your answer. 11

12 Gravitation PROBLEM SET (answers at end of problem set) 1) 2) What is the force of gravity between earth and the moon? The earth s mass is 5.98 x kg, the distance from the earth to the moon is 3.90 x 10 8 m. The mass of the moon is 7.30 x kg. 3) Is the earth s gravitational force on the sun larger than, smaller than or equal to the sun s gravitational force on the earth? Explain. 4) 5) A space rock has a mass of 2.50 kg. It is 1,250 m from an asteroid. If the force of gravity is N between them, what is the mass of the asteroid? 6) A 65,700 kg celestial body experiences a 1.25 N force of attraction between it and a 75,500 kg space rock. How far apart are they? 12

13 Gravitation PROBLEM SET (answers at end of problem set) 7) The gravitational force of a star on orbiting plant 1 of F 1. Planet 2, which is twice as massive as Planet 1 and orbits at twice the distance from the star, experiences gravitational force F 2.What is the ratio of F 2 /F 1? 8) You weigh 458 N on earth, but you are on Mars. Here s some data on Mars: radius = 3.38 x 10 6 m, (a) How much do you weigh on Mars? mass = 6.42 x kg. (b) What is the acceleration of gravity on Mars? (c) If you drop a 3.50 kg rock above the surface of Mars and it falls a distance of 1.20 m, how fast will it be going just before it hits the surface? 9) The acceleration due to gravity at the surface of Planet X is 20 m/s 2. The radius and the mass of Planet Z are twice those of Planet X. What is g on Planet Z? 10) Determine the speed of the Hubble Space Telescope orbiting the earth at a height of 598 km above the surface (R E = 6.38x10 6 m, M E = 5.98 x kg). b. What would the speed be if the radius above the earth was cut in half? 13

14 Gravitation PROBLEM SET (answers at end of problem set) 11) Satellite A orbits a planet with a speed of 10,000 m/s. Satellite B, orbiting at the same distance from the center of the planet is twice as massive as Satellite A. What is the speed of Satellite B? 12) Spaceman Spiff orbits planet X in his spaceship. To remain in orbit at 421 km from the planets center, he must maintain a speed of 80 m/s. What is the mass of planet X? 13) (challenge) A digital TV satellite is placed in geosynchronous orbit around Earth, so it is always in the same spot in the sky. a) Using the fact that the satellite will have the same period of revolution as Earth, calculate the radius of its orbit. b) What is the ratio of the radius of this orbit to the radius of the Earth? c) If the mass of the satellite were to double, would the radius of the satellite s orbit be larger, smaller, or the same? Why? 14) It has been proposed that future space stations create artificial gravity by rotating around an axis. a) How would this work? Explain b) Would the artificial gravity be equally effective throughout the space station? If not, where in the space stations would the residents want to live and work? 14

15 Gravitation PROBLEM SET (answers at end of problem set) 15) A space station was established far from the gravitational field of Earth. Extended stays in zero gravity are not healthy for human beings. Thus, for the comfort of the astronauts, the station is rotated so that the astronauts feel there is an internal gravity. The rotation speed is such that the apparent acceleration of gravity is 9.8 m/s 2. The direction of rotation is counter-clockwise. a) If the radius of the station is 80 m, what is its tangential speed, v? b) Draw vectors representing the astronaut s velocity and acceleration on the picture above. c) Draw a free body diagram for the astronaut on the dot below for the position she is in the picture above. d) Is the astronaut exerting a force on the space station for the position she is in? If so, calculate its magnitude. Her mass m = 65 kg. e) The astronaut drops a ball, which appears to accelerate to the floor, (see picture) at 9.8 m/s 2 i) Draw the velocity and acceleration vectors for the ball while it is in the air on ball in the air on the picture to the right. ii) What force(s) are acting on the ball while it is in the air? iii) Draw the acceleration and velocity vectors after the ball hits the floor and comes to rest on the ball on the floor in the picture to the right. iv) What force(s) act on the ball after it hits the ground? v) Why does the ball land at your feet? 15

16 Answers to PROBLEM SET Circular Motion: s s 3. V 3 > V 2 = V 1 4. V 1, V 2 = 0.38 m/s V 3 = m/s 5. straight up 6. T 3 > T 1 = T 4 > T 2 7. a) 7.54 m/s b) 94.7 m/s 2 c) 14.2 N 8. a) 1023 m/s b) m/s 2 c) 2.0 x N m/s c) to stay in circle must be inward force d) when reach f smax e) f) g) 11. a) static friction b) 1430 N c) a) 0.38 m/s b) a) 7.23 m/s b) 12.4 m/s Fn is less than Fg 15. Greater Gravitation Problems: 1) 2) 1.9 x N 3) Equal 4) a) b) smaller 5) 2.11x10 9 kg 6) 0.51m 7) ½ 8) a) 175 N b) 3.75 m/s 2 c) 4.8 m/s 9) 10 m/s 2 10) a) 7560 m/s b) 7729 m/s 11) mass doesn t matter 12) 4.0 x10 19 kg 13) a) 4.22 x 10 7 m b) 6.6/1 c) same 14) a) rotating hollow space station would have a centripetal force that would feel like a normal force b) Varies with r so inside the outer shell 15) a) 28 m/s d) yes 637 N e) i) none iv) Fn v) rotational speed is constant so the ball would travel at about same speed as your feet (but you are rotating) bottom & 20 See online answers 21. a) b) 10 m/s c) 30 N 22. b) 2.42 m/s c) 1.5 m 23. a) 2.05 m/s c) 2.67 N d) J e) decrease radius or increase initial height. 16

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