Chapter2 M OTION OUTLINE Describing Motion 2.1 Speed 2.2 Vectors 2.3 Acceleration 2.4 Distance, Time, and Acceleration Acceleration of Gravity 2.5 Free Fall 2.6 Air Resistence Force and Motion 2.7 First Law of Motion 2.8 Mass 2.9 Second Law of Motion 2.10 Mass and Weight 2.11 Third Law of Motion Gravitation 2.12 Circular Motion 2.13 Newton s Law of Gravity 2.14 Artificial Satellites GOALS 1. Distinguish between instantaneous and average speeds. 2. Use the formula v = d/t to solve problems that involve distance, time, and speed. 3. Distinguish between scalar and vector quantities and give several examples of each. 4. Use the Pythagorean theorem to add two vector quantities of the same kind that act as right angles to each other. 5. Define acceleration and find the acceleration of an object whose speed is changing. 6. Use the formula v 1 = v 2 + at to solve problems that involve speed, acceleration, and time. 2 7. Use the formula d = v1t + ½ at to solve problems that involve distance, time, speed, and acceleration. 8. Explain what is meant by the acceleration of gravity. 9. Describe the effect of air resistance on falling objects. 10. Separate the velocity of an object into vertical and horizontal components in order to determine its motion. 11. Define force and indicate its relationship to the first law of motion. 12. Discuss the significance of the second law of motion, F = ma. 13. Distinguish between mass and weight and find the weight of an object of given mass. 14. Use the third law of motion to relate action and reaction forces. 15. Explain the significance of centripetal force in motion along a curved path. 16. Relate the centripetal force on an object moving in a circle to its mass, speed, and the radius of the circle. 17. State Newton's law of gravity and describe how gravitational forces vary with distance. 18. Account for the ability of a satellite to orbit the earth without either falling to the ground or flying off into space. 19. Define escape speed. 14
CHAPTER SUMMARY This chapter discusses the concept of motion, both straight-line and circular. Motion is described in terms of speed, velocity, and acceleration. Scalar quantities are differentiated from vector quantities, and the use of vectors to represent vector quantities is presented. The downward acceleration due to gravity is shown to influence the motion of objects in free fall near the earth's surface. The behavior of moving bodies is summarized by Newton's three laws of motion. The concept of inertia is developed in the first law of motion, and the relationship between force, mass, and acceleration is operationally defined by the second law of motion. Weight is defined as a force, and the relationship between weight and mass is described operationally in terms of the second law. The concept that forces come in pairs consisting of an action force and a reaction force is developed in the third law of motion. Circular motion is shown to be the result of centripetal force acting upon a moving body. CHAPTER OUTLINE 2-1. Speed A. The speed of a moving object is the rate at which it covers distance: where v = speed, d = distance, and t = time. B. Average speed is the total distance traveled by an object divided by the time taken to travel that distance. C. Instantaneous speed is an object's speed at a given instant of time. 2-2. Vectors A. The magnitude of a quantity tells how large the quantity is. B. There are two types of quantities: 1. Scalar quantities have magnitude only. 2. Vector quantities have both magnitude and direction. C. Velocity is a vector quantity that includes both speed and direction. D. A vector can be represented by an arrowhead line whose length is proportional to the magnitude of some quantity and whose direction is that of the quantity. 2-3. Acceleration A. The acceleration of an object is the rate of change of its velocity and is a vector quantity. B. For straight-line motion, average acceleration is the rate of change of speed: 15
where a = acceleration, t = time, v = final speed, and v = initial speed. 2-4. Distance, Time, and Acceleration f A. Speed and acceleration are defined quantities. B. Relating speed and acceleration to each other and to time (a measurable quantity) allows the formation of equations that can answer certain questions involving distance, time, and acceleration. 2-5. Free Fall A. The acceleration due to gravity (g) for objects in free fall at the earth's surface is 9.8 2 m/s. 2 B. An object thrown horizontally has a vertical acceleration of 9.8 m/s and a constant horizontal velocity; its trajectory is a curved path. 2 C. An object thrown upward falls back to earth at an acceleration of 9.8 m/s and returns to its starting point with the same speed at which it was thrown. D. An object thrown downward has a final speed that is the sum of its original speed plus the increase in speed due to the acceleration of gravity. E. An object thrown upward at an angle to the ground follows a curved path called a parabola. 2-6. Air Resistance A. Air resistance prevents a falling object from reaching the full speed produced by the acceleration of gravity. B. Air resistance increases with the speed of a moving object. C. Terminal speed is the maximum speed a falling object obtains when the force due to downward acceleration of gravity is balanced by the upward force of air resistance. D. The terminal speed for any object depends on the object's size, shape, and mass. 2-7. First Law of Motion A. Isaac Newton formulated the three laws of motion. B. The first law of motion states: If no net force acts on it, an object at rest remains at rest and an object in motion remains in motion at a constant velocity. C. A force is any influence that can cause an object to be accelerated. D. Every acceleration can be traced to the action of a force. i 16
2-8. Mass A. Inertia is the apparent resistance an object offers to any change in its state of rest or motion. B. The mass of a body is the property of matter that manifests itself as inertia. 1. Mass may be thought of as a quantity of matter. 2. The greater an object's mass, the greater the object's inertia. C. The SI unit for mass is the kilogram (kg). 2-9. Second Law of Motion A. Newton's second law of motion states: The net force on an object equals the product of the mass and the acceleration of the object. The direction of the force is the same as that of the acceleration. B. The second law of motion can be expressed in either of these ways: where a = acceleration, F = force, and m = mass, or where F = force, m = mass, and a = acceleration. C. The SI unit of force is the newton (N): 1 newton = 1 N = 1 (kg)(m/s 2 ) D. The pound (lb) is the unit of force in the British system of measurement: 2-10. Mass and Weight 1 lb = 4.45 N (1 N = 0.225 lb) A. The weight of an object is the force with which gravity pulls it toward the earth: w = mg where w = weight, m = mass, and g = acceleration of gravity (9.8 m/s ). 2 B. In the SI, mass rather than weight is normally specified. C. On earth, the weight of an object (but not its mass) can vary because the pull of gravity is not the same everywhere on earth. 17
2-11. Third Law of Motion A. The third law of motion states: When one object exerts a force on a second object, the second object exerts an equal force in the opposite direction on the first object. B. No force ever occurs singly; for every action force there is an equal but opposite reaction force: 1. The action force is the force the first object exerts on the second. 2. The reaction force is the force the second object exerts on the first. 2-12. Circular Motion A. Centripetal force is the inward force exerted on an object to keep it moving in a curved path: where F c = centripetal force, m = mass, v = speed, and r = radius of circular path. 1. The greater the object's mass, the greater the centripetal force. 2. The faster the object moves, the greater the centripetal force. 3. The smaller the circle (curved path) of the object, the greater the centripetal force. B. Highway curves are banked so that the horizontal component of the reaction force the tilted road exerts on the car provides the centripetal force needed to keep the car from skidding. 2-13. Newton's Law of Gravity A. Newton used Galileo's work on falling bodies and Kepler's laws of planetary motion to develop his law of gravity. B. Newton's law of gravity states: Every object in the universe attracts every other object with a force proportional to both of their masses and inversely proportional to the square of the distance between them. 11 2 2 where F = force, G = gravitational constant (6.670 x 10 N m /kg ), m 1 = mass of object 1, m 2 = mass of object 2, and R = distance between objects. C. The force of gravitation drops off rapidly as the distance between two objects increases. D. The center of mass of an object is the point where the mass of the object appears to be concentrated; for a uniform sphere, the center of mass is its geometric center. 18
2-14. Artificial Satellites A. The world's first artificial satellite was Sputnik I, launched in 1957 by the Soviet Union. B. A satellite in a geostationary orbit remains in place indefinitely over a particular location on the earth. C. To maintain a circular orbit, the centripetal force on the satellite must be equal to the gravitational force on the satellite. D. The escape speed is the speed required by an object to leave the gravitational influence of an astronomical body; for earth, this speed is about 40,000 km/h. KEY TERMS AND CONCEPTS The questions in this section will help you review the key terms and concepts from Chapter 2. Multiple Choice Circle the best answer for each of the following questions. 1. The rate at which an object travels a certain distance is known as a. velocity b. speed c. momentum d. acceleration 2. While driving between Pittsburgh and Chicago, the driver of the car looks at the car's speedometer and notes a speed of 50 miles/hour. The speedometer is indicating the car's a. average speed b. velocity c. terminal speed d. instantaneous speed 3. Which one of the following represents a vector quantity? a. 750 miles/h b. 60 cycles/second c. 220 volts d. 55 km/h toward the north 4. The acceleration due to gravity at the earth's surface is a. 32 m/s 2 b. 9.8 m/s 2 c. 1.6 m/s 2 d. 5.4 m/s 2 19
5. Newton's first law of motion deals with a. force b. acceleration c. weight d. inertia 6. A tennis player hits a tennis ball with a tennis racquet. If the force supplied by the racquet to the ball is the action force, what would be the reaction force? a. the force transmitted from the racquet to the tennis player's arm b. the resistance of air molecules on the tennis ball c. the force exerted by the tennis ball on the racquet d. the force exerted by the tennis ball on the opposing player's racquet 7. A bowling ball and a marble are dropped at the same time in a vacuum chamber. Which one of the following statements correctly describes the results? a. The bowling ball will land first. b. The acceleration of the bowling ball is greater than that of the marble. c. Both have the same speed upon landing. d. The force of impact will be the same for both the bowling ball and the marble. Refer to the above drawing to answer questions 8 and 9. A ball is thrown upward and is shown at several positions along its trajectory. 8. The speed of the ball at point C is a. greater than at point A b. at its maximum c. zero d. 9.8 m/s 20
9. The time interval between each change in position of the ball is a. greatest between points F and G b. at a minimum between points B and C and points C and D c. shortest between points F and G d. approximately equal Refer to the above drawing to answer questions 10 through 12. 10. Given that F c = centripetal force, v = speed, m = mass, and r = radius, doubling the value of m would a. reduce the value of F c by 1/2 b. double the value of F c c. quadruple the value for F c d. have no effect on the value of F c 11. Doubling the value of v would a. reduce the value of F c by 1/2 b. quadruple the value of F c c. double the value of F c d. have no effect on the value of F c 12. Doubling the value of r would a. double the value of F c b. reduce the value of F c by 1/2 c. quadruple the value of F c d. have no effect on the value of F c 21
Refer to the above drawing to answer questions 13 and 14. 13. The string a. provides gravitational force b. represents the force of inertia c. represents the center of mass d. provides centripetal force 14. The centripetal force vector F c is always a. tangent to the circle b. directed toward the center of the circle c. directed vertically toward the earth d. directed away from the center of the circle Refer to the above drawing to answer question 15. The car was originally at rest. 15. The best explanation for the position of the passengers within the car is a. the car's sudden start throws the passengers back into their seats b. the passenger's inertia tends to keep them in their original positions as the car (and the seats) move forward c. the car's forward acceleration is balanced by the opposing force of the passengers moving backward in their seats d. the car's seats broke at the moment of acceleration 22
True or False Decide whether each statement is true or false. If false, briefly state why it is false or correct the statement to make it true. The first question has been completed as an example. FALSE 1. Speed is an example of a vector quantity. The speed at which an object moves reveals how fast the object is going, not its direction. 2. The SI unit of force is the newton. 3. Scalar quantities require only a number and a unit to be completely specified. 4. Assuming there is no wind or air friction, a ball thrown upward at a certain speed will return to its starting point with the same speed. 5. The speed or direction of any moving object stays unchanged unless a net force acts upon the object. 6. Inertia is the force that resists any change in an object s state of rest or motion. 7. The direction of acceleration of a moving object is always the same as the direction of the net force(s) acting on the object. 8. Your weight remains constant no matter where you are on the earth's surface. 9. The force that has to be applied to an object to make it move in a curved path is called centrifugal force. 10. If the distance between a planet and its sun were tripled, the gravitational force applied on the planet by its sun would drop to one-ninth of its original amount. 23
Fill in the Blank Complete the following statements by filling in the missing term. 1. Speed is equal to distance divided by. 2. Distance is equal to the product of and time. 3. is equal to distance divided by speed. 4. According to Newton's second law of motion, acceleration is equal to divided by mass. 5. Force is equal to mass times. 6. is equal to mass times the acceleration of gravity. 7. The curved path that an object takes when it is thrown upward at an angle to the ground is called a(n). 8. A(n) may be represented by an arrowed line that represents the magnitude and direction of a quantity. 9. The is the British system unit of force. 10. The vector quantity that includes both speed and direction is called. 24
Matching Match the terms on the left with their definitions on the right. 1. force 2. mass 3. Newton s law of gravity 4. inertia 5. vector quantity 6. Newton s first law of motion 7. newton 8. centripetal force 9. scalar quantity 10. Newton s third law of motion a. inward force on an object moving in a curved path b. resistance an object offers to any change in its state of rest or motion c. states that, if no net force acts on it, every object continues in its state of rest or uniform motion in a straight line. d. states that when one object exerts a force on a second object, the second object exerts an equal but opposite force on the first. e. has both magnitude and direction f. states that every object in the universe attracts every other object with a force directly proportional to both their masses and inversely proportional to the square of the distance between them. g. any influence that can cause an object to be accelerated. h. has magnitude only i. SI unit for power j. quantity of matter in an object SOLVED PROBLEMS Study the following solved example problems as they will provide insight into solving the problems listed at the end of Chapter 2 in The Physical Universe. Review the mathematics refresher in the text if you are unfamiliar with the basic mathematical operations presented in these examples. Example 2-1 Oh no! An enraged bull 48 m due north has spotted you in an open field and has decided to charge you. Your only chance of escape is to climb up a tree 24 m due south of you. The bull makes its charge at the exact moment you run toward the tree. If the bull is running toward you at 12 m/s and you run at 6 m/s, will you reach the safety of the tree before the bull gores you? Solution Both you and the pursuing bull are moving toward the tree. Because the bull was 48 m north of you, it has to cover a distance of 72 m to reach the tree. 25
To determine how long it will take the bull to reach the tree, use the formula for time. Substituting for d and v and dividing, we get Note that the m units cancel, leaving the answer expressed in units of time (s). Now let s find out how long it takes you to reach the tree and safety. Assuming you are good at climbing trees, you are safe. Example 2-2 A 1240-kg car goes from 16 m/s to 26 m/s in 20 s. What is the average force acting upon it? Solution To determine the average force acting on the car, we use Newton's second law of motion expressed as: F = ma We are given the car's mass (1240 kg), but not its acceleration. To determine the car's acceleration, we can use the formula for acceleration: Substituting for v, v, and t, we get: f i 26
We can now substitute the quantities for mass and acceleration into the formula for force and solve the problem. Example 2-3 2 2 F = ma = (1240 kg)(0.5 m/s ) = 620 kg m/s = 620 N A traditional Scottish highland game is the hammer throw in which the athlete spins the hammer in a circular motion building momentum prior to releasing the hammer. If the head of the hammer weighs 6.7 kg and is moving at 6 m/s in a circular arc of 1.2 m radius, how much force must the athlete exert on the handle of the hammer to prevent it from flying out of his hands? Ignore the mass of the hammer s light wood handle. This is a centripetal force problem, and to solve it, we use the formula for centripetal force. Substituting the quantities given to us in the problem, we get WEB LINKS This interactive web site illustrates the concept of velocity composition by showing the velocity vectors relevant to the motion of a boat on a river: http://www.glenbrook.k12.il.us/gbssci/phys/shwave/rboat.html Investigate the relationship between force and motion by guiding a race car around an oval track in the least number of moves: http://www.glenbrook.k12.il.us/gbssci/phys/shwave/racetrack.html Control the trajectories of a cannonball at this interactive web site: http://jersey.uoregon.edu/vlab/cannon/ Study projectile orbits and satellite orbits at this interactive web site: http://www.phy.ntnu.edu.tw/java/projectileorbit/projectileorbit.html 27
ANSWER KEY Multiple Choice 1. b 2. d 3. d 4. b 5. d 6. c 7. c 8. c 9. d 10. b 11. b 12. b 13. d 14. b 15. b True or False 1. False. See explanatory notes given in the example. 2. True 3. True 4. True 5. True 6. False. Inertia is not an actual force. A force is any influence that can change the speed or direction of motion of an object. Inertia is the resistance offered by an object to any change in its state of rest or motion. 7. True 8. False. The pull of gravity is not exactly the same everywhere on earth; therefore, a person's weight will vary depending upon the person's location. 9. False. The force that has to be applied to make an object move in a curved path is called centripetal force. 10. True Fill in the Blank 1. time 6. Weight 2. speed 7. parabola 3. Time 8. vector 4. force 9. pound 5. acceleration 10. velocity Matching 1. g 2. j 3. f 4. b 5. e 6. c 7. i 8. a 9. h 10. d 28