Forces and Motion: UNIT 1 Dynamics 2 OVERALL EXPECTATIONS UNIT CONTENTS ANALYZE, predict, and explain the motion of selected objects in vertical, horizontal, and inclined planes. CHAPTER 1 Fundamentals of Dynamics INVESTIGATE, represent, and analyze motion and CHAPTER 2 Dynamics in Two forces in linear, projectile, and circular motion. Dimensions RELATE your understanding of dynamics to the CHAPTER 3 Planetary and Satellite development and use of motion technologies. Dynamics S pectators are mesmerized by trapeze artists making perfectly timed releases, gliding through gracefu l arcs, and intersecting the paths of their partners. An error in timing and a graceful arc could become a trajectory of panic. Trapeze artists know that tiny differences in height, velocity, and timing are critical. Swinging from a trapeze, the performer forces his body from its natural straight- line path. Gliding freely through the air, he is subject only to gravity. Then, the outstretched hands of his partner make contact, and the performer is acutely aware of the forces that change his speed and direction. In this unit, you will explore the relationship between motion and the forces that cause it and investigate how different perspectives of the same motion are related. You will learn how to analyze forces and motion, not only in a straight line, but also in circular paths, in parabolic trajectories, and on inclined surfaces. You will discover how the motion of planets and satellites is caused, described, and analyzed. UNIT PROJECT PREP Refer to pages 126–127 before beginning this unit. In the unit project, you will design and build a working catapult to launch small objects through the air. (cid:2) What launching devices have you used, watched, or read about? How do they develop and control the force needed to propel an object? (cid:2) What projectiles have you launched? How do you direct their flight so that they reach a maximum height or stay in the air for the longest possible time? 3 C H A P T E R 1 Fundamentals of Dynamics CHAPTER CONTENTS Multi-Lab Thinking Physics 5 1.1 Inertia and Frames of Reference 6 Investigation 1-A Measuring Inertial Mass 8 1.2 Analyzing Motion 15 1.3 Vertical Motion 27 Investigation 1-B Atwood’s Machine 34 1.4 Motion along an Incline 46 PREREQUISITE CONCEPTS AND SKILLS (cid:2) Using the kinematic equations for uniformly accelerated motion. H ow many times have you heard the saying, “It all depends on your perspective”? The photographers who took the two pictures of the roller coaster shown here certainly had different perspectives. When you are on a roller coaster, the world looks and feels very different than it does when you are observing the motion from a distance. Now imagine doing a physics experiment from these two perspectives, studying the motion of a pendulum, for example. Your results would definitely depend on your perspective or frame of reference. You can describe motion from any frame of reference, but some frames of reference simplify the process of describing the motion and the laws that determine that motion. In previous courses, you learned techniques for measuring and describing motion, and you studied and applied the laws of motion. In this chapter, you will study in more detail how to choose and define frames of reference. Then, you will extend your knowledge of the dynamics of motion in a straight line. 4 MHR • Unit 1 Forces and Motion: Dynamics TARGET SKILLS Thinking Physics M U L T I Predicting L A B Identifying variables Analyzing and interpreting Suspended Spring Analyze and Conclude Tape a plastic cup to one end of a short 1. Describe the motion of the cup and the section of a large-diameter spring, such as lower end of the spring. Compare the a Slinky™. Hold the other end of the spring motion to your prediction and describe high enough so that the plastic cup is at least any differences. 1 m above the floor. Before you 2. Is it possible for any unsupported object release the spring, predict the to be suspended in midair for any length exact motion of the cup of time? Create a detailed explanation to from the instant that it is account for the behaviour of the cup at the released until the moment moment at which you released the top of that it hits the floor. While the spring. your partner watches the 3. Athletes and dancers sometimes seem to cup closely from a kneel- be momentarily suspended in the air. ing position, release the How might the motion of these athletes top of the spring. Observe be related to the spring’s movement in the motion of the cup. this lab? Thought Experiments 2. A golf pro drives a ball through the air. What force(s) is/are acting on the golf ball Without discussing the following questions for the entirety of its flight? with anyone else, write down your answers. (a) force of gravity only 1. Student A and A B (b) force of gravity and the force of Student B sit in the “hit” identical office chairs facing (c) force of gravity and the force of air resistance each other, as illustrated. (d) force of gravity, the force of the “hit,” Student A, who and the force of air resistance is heavier than Student B, suddenly push- 3. A photographer es with his feet, causing both chairs to accidentally drops move. Which of the following occurs? a camera out of a A B C D (a) Neither student applies a force to the small airplane as other. it flies horizontally. As seen from the (b) A exerts a force that is applied to B, ground, which path would the camera but A experiences no force. most closely follow as it fell? (c) Each student applies a force to the other, but A exerts the larger force. Analyze and Conclude (d) The students exert the same amount Tally the class results. As a class, discuss the of force on each other. answers to the questions. Chapter 1 Fundamentals of Dynamics • MHR 5 Inertia and Frames 1.1 of Reference Imagine watching a bowling ball sitting still in the rack. Nothing SECTION EXPECTATIONS moves; the ball remains totally at rest until someone picks it up • Describe and distinguish and hurls it down the alley. Galileo Galilei (1564–1642) and later between inertial and non- Sir Isaac Newton (1642–1727) attributed this behaviour to the inertial frames of reference. property of matter now called inertia, meaning resistance to changes in motion. Stationary objects such as the bowling ball • Define and describe the remain motionless due to their inertia. concept and units of mass. Now picture a bowling ball rumbling down the alley. • Investigate and analyze Experience tells you that the ball might change direction and, if linear motion, using vectors, the alley was long enough, it would slow down and eventually graphs, and free-body stop. Galileo realized that these changes in motion were due to diagrams. factors that interfere with the ball’s “natural” motion. Hundreds of years of experiments and observations clearly show that Galileo KEY was correct. Moving objects continue moving in the same direc- TERMS tion, at the same speed, due to their inertia, unless some external • inertia force interferes with their motion. • inertial mass • gravitational mass • coordinate system • frame of reference • inertial frame of reference • non-inertial frame of reference • fictitious force Figure 1.1 You assume that an inanimate object such as a bowling ball will remain stationary until someone exerts a force on it. Galileo and Newton realized that this “lack of motion” is a very important property of matter. Analyzing Forces Newton refined and extended Galileo’s ideas about inertia and straight-line motion at constant speed — now called “uniform motion.” NEWTON’S FIRST LAW: THE LAW OF INERTIA An object at rest or in uniform motion will remain at rest or in uniform motion unless acted on by an external force. 6 MHR • Unit 1 Forces and Motion: Dynamics Newton’s first law states that a force is required to change an LANGUAGE LINK object’s uniform motion or velocity. Newton’s second law then permits you to determine how great a force is needed in order to The Latin root of inertiameans change an object’s velocity by a given amount. Recalling that “sluggish” or “inactive.” An inertial acceleration is defined as the change in velocity, you can state guidance systemrelies on a gyro- (cid:2) Newton’s second law by saying, “The net force (F ) required to scope, a “sluggish” mechanical device (cid:2) accelerate an object of mass m by an amount (a ) is the product that resists a change in the direction of motion. What does this suggest of the mass and acceleration.” about the chemical properties of an inert gas? NEWTON’S SECOND LAW The word equation for Newton’s second law is: Net force is the product of mass and acceleration. (cid:2) (cid:2) F = ma QuantitySymbol SI unit (cid:2) force F N (newtons) mass m kg (kilograms) (cid:2) m acceleration a (metres per second s2 squared) Unit analysis (cid:2) (cid:3) (mass)(acceleration)= (kilogram) metres kgm = kg·m = N second2 s2 s2 (cid:2) Note: The force (F ) in Newton’s second law refers to the vector sum of all of the forces acting on the object. Inertial Mass When you compare the two laws of motion, you discover that the first law identifies inertia as the property of matter that resists a change in its motion; that is, it resists acceleration. The second law gives a quantitative method of finding acceleration, but it does not seem to mention inertia. Instead, the second law indicates that the property that relates force and acceleration is mass. Actually, the mass (m) used in the second law is correctly described as the inertial mass of the object, the property that resists a change in motion. As you know, matter has another prop- erty — it experiences a gravitational attractive force. Physicists refer to this property of matter as its gravitational mass. Physicists never assume that two seemingly different properties are related without thoroughly studying them. In the next investigation, you will examine the relationship between inertial mass and gravita- tional mass. Chapter 1 Fundamentals of Dynamics • MHR 7 I N V E S T I G A T I O N 1-A TARGET SKILLS Hypothesizing Measuring Inertial Mass Performing and recording Analyzing and interpreting Problem 4. Add unit masses one at a time and measure Is there a direct relationship between an object’s the acceleration several times after each inertial mass and its gravitational mass? addition. Average your results. 5. Graph the acceleration versus the number of Hypothesis unit inertial masses on the cart. Formulate an hypothesis about the relationship between inertial mass and its gravitational mass. 6. Remove the unit masses from the cart and replace them with the unknown mass, then Equipment measure the acceleration of the cart. (cid:2) dynamics cart 7. Use the graph to find the inertial mass of the (cid:2) pulley and string unknown mass (in unit inertial masses). (cid:2) laboratory balance 8. Find the gravitational mass of one unit of (cid:2) standard mass (about 500 g) inertial mass, using a laboratory balance. (cid:2) metre stick and stopwatch ormotion sensor 9. Add a second scale to the horizontal axis of (cid:2) unit masses (six identical objects, such as small your graph, using standard gravitational mass C-clamps) units (kilograms). (cid:2) unknown mass (measuring between one and six unit masses, such as a stone) 10. Use the second scale on the graph to predict the gravitational mass of the unknown mass. Procedure 11. Verify your prediction: Find the unknown’s 1. Arrange the pulley, string, standard mass, gravitational mass on a laboratory balance. and dynamics cart on a table, as illustrated. Analyze and Conclude 1. Based on your data, are inertial and dynamics gravitational masses equal, proportional, cart pulley or independent? 2. Does your graph fit a linear, inverse, expo- nential, or radical relationship? Write the relationship as a proportion (a ∝?). 3. Write Newton’s second law. Solve the standard expression for acceleration. Compare this mass expression to your answer to question 2. What inferences can you make? 2. Set up your measuring instruments to deter- 4. Extrapolate your graph back to the vertical mine the acceleration of the cart when it is axis. What is the significance of the point at pulled by the falling standard mass. Find which your graph now crosses the axis? the acceleration directly by using computer software, or calculate it from measurements 5. Verify the relationship you identified in of displacement and time. question 2 by using curve-straightening techniques (see Skill Set 4, Mathematical 3. Measure the acceleration of the empty cart. Modelling and Curve Straightening). Write a specific equation for the line in your graph. 8 MHR • Unit 1 Forces and Motion: Dynamics Over many years of observations and investigations, physicists concluded that inertial mass and gravitational mass were two different manifestations of the same property of matter. Therefore, when you write m for mass, you do not have to specify what type of mass it is. Action-Reaction Forces Newton’s first and second laws are sufficient for explaining and predicting motion in many situations. However, you will discover that, in some cases, you will need Newton’s third law. Unlike the first two laws that focus on the forces acting on one object, Newton’s third law considers two objects exerting forces on each other. For example, when you push on a wall, you can feel the wall pushing back on you. Newton’s third law states that this condition always exists — when one object exerts a force on another, the second force always exerts a force on the first. The third law is sometimes called the “law of action-reaction forces.” NEWTON’S THIRD LAW For every action force on an object (B) due to another object (A), there is a reaction force, equal in magnitude but opposite in direction, on object A, due to object B. (cid:2) (cid:2) F = −F AonB BonA To avoid confusion, be sure to note that the forces described in Newton’s third law refer to two different objects. When you apply Newton’s second law to an object, you consider only one of these forces — the force that acts on the object. You do not include any forces that the object itself exerts on something else. If this concept is clear to you, you will be able to solve the “horse-cart paradox” described below. Conceptual Problem • The famous horse-cart paradox asks, “If the cart is pulling on the horse with a force that is equal in magnitude and opposite in direction to the force that the horse is exerting on the cart, how can the horse make the cart move?” Discuss the answer with a classmate, then write a clear explanation of the paradox. Chapter 1 Fundamentals of Dynamics • MHR 9 TARGET SKILLS Bend a Wall Q U I C K Initiating and planning L A B Bend a Wall Performing and recording Analyzing and interpreting Sometimes it might not seem as though an Analyze and Conclude object on which you are pushing is exhibiting 1. Calculate the extent of the movement (s) — any type of motion. However, the proper appa- or how much the wall “bent” — using the ratus might detect some motion. Prove that you formula s = rS. can move — or at least, bend — a wall. 2R 2. If other surfaces behave as the wall does, CAUTION Do not look into the laser. list other situations in which an apparently Glue a small mirror to a 5 cm T-head dissect- inflexible surface or object is probably ing pin. Put a textbook on a stool beside the moving slightly to generate a resisting or wall that you will attempt to bend. Place the supporting force. pin-mirror assembly on the edge of the textbook. 3. Do your observations “prove” that the wall As shown in the diagram, attach a metre stick to bent? Suppose a literal-minded observer the wall with putty or modelling clay and rest questioned your results by claiming that you the other end on the pin-mirror assembly. The did not actually see the wall bend, but that pin-mirror should act as a roller, so that any you actually observed movement of the laser movement of the metre stick turns the mirror spot. How would you counter this objection? slightly. Place a laser pointer so that its beam reflects off the mirror and onto the opposite 4. Is it scientifically acceptable to use a mathe- wall. Prepare a linear scale on a sheet of paper matical formula, such as the one above, and fasten it to the opposite wall, so that you without having derived or proved it? Justify can make the required measurements. your response. 5. If you have studied the arc length formula in mathematics, try to derive the formula above. opposite wall wall (Hint: Use the fact that the angular displace- ment of the laser beam is actually twice the poster putty angular displacement of the mirror.) laser rod or metre stick Apply and Extend scale 6. Imagine that you are explaining this experi- dissecting ment to a friend who has not yet taken a pin physics course. You tell your friend that S “When I pushed on the wall, the wall R textbook mirror pushed back on me.” Your friend says, “That’s silly. Walls don’t push on people.” Use the laws of physics to justify your original statement. Push hard on the wall near the metre stick and 7. Why is it logical to expect that a wall will observe the deflection of the laser spot. Measure move when you push on it? (cid:2) the radius of the pin (r) 8. Dentists sometimes check the health of your (cid:2) the deflection of the laser spot (S) teeth and gums by measuring tooth mobility. (cid:2) the distance from the mirror to the opposite Design an apparatus that could be used to wall (R) measure tooth mobility. 10 MHR • Unit 1 Forces and Motion: Dynamics