Chapter 4: Forces and Newton's Laws of Motion (Part 1) Tuesday, September 17, 2013

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1 Chapter 4: Forces and Newton's Laws of Motion (Part 1) Tuesday, September 17, :00 PM In Chapters 2 and 3 of this course the emphasis is on kinematics, the mathematical description of motion. In Chapter 4, the heart of the course, we shift the emphasis to explaining why changes in motions occur. This branch of mechanics is called dynamics. Together, what we have learned and what we will learn, kinematics and dynamics, form the foundation of Newtonian mechanics. "Everything happens for a reason," as they say. In Newtonian mechanics the reason always has to do with a force. Clarifying these vague statements is the goal of this chapter. One perspective on these vague statements is causality, one of the fundamental principles of science: Each effect has a cause. But what is it about motion that is caused by a force? For Aristotle, one of the greatest thinkers of the Ancient Greek era, the natural state of an object is to be at rest. Any deviation from rest needed explaining in terms of some cause. This seems sensible; in our experience, objects don't suddenly fly around for no reason. Things need to be pushed or pulled in order to get started moving, and if you push something and then stop pushing, then our experience is that it eventually stops. So it's natural that Aristotle, and his followers for nearly two millennia, came to accept this as the truth about motion. And it is true, as far as it goes; but there are deeper truths, and it took Kepler, Galileo, Newton, and others to tease out richer understandings. One of Newton's great advances was to connect forces to acceleration, whereas previously scientists tried to connect forces to velocity. It's natural to say that the harder you push something the faster it goes, and if you stop pushing it slows down and stops. Newton built on the work of Galileo, who used his imagination to consider motions in an ideal world without friction. In the absence of friction, Galileo reckoned that an object moving at a constant speed in a straight line would continue moving in the same straight line at the same constant speed. Thus, Galileo argued, motion in a straight line at a constant speed is just as "natural" as a state of rest. He shifted the central question from, "What to humans do to create motion?" to the much more fertile question, "What are all the influences on a moving object?" This led to the modern concept of force. This fundamental shift in understanding helped to change our conception of "force" from something exerted only by humans (and other animate creatures), to a more general kind of phenomenon. It was gradually realized that inanimate objects could also exert forces on other objects. This was a great advance, but perhaps Galileo's greatest advance was to promote the idea that understanding the natural world required mathematical and experimental means. At the time, it was accepted that the highest form of natural philosophy involved careful reading of Aristotle and commentaries and discussions of his works. Ch4A Page 1

2 Galileo declared that the world was written in the language of mathematics, and to understand it we need to perform experiments and make measurements to collect numerical data above all else, not rely on the opinions of authorities, no matter how great. Therefore many people consider Galileo the founder of modern science. (The full story is more complicated than is possible to summarize in these brief notes, but Galileo does stand out as a decisive figure.) We'll begin by examining some critical scenarios to determine your prior knowledge; then we'll discuss the examples, and bring in the key concepts of the chapter. Almost everyone entering a first-year university physics course has many misconceptions about Newton's laws; this is normal and natural, because Newton's laws are extremely non-intuitive. Remember that it took the world's greatest scientific minds about 2000 years to correct the misconceptions held by virtually every intelligent person, from the great Aristotle onwards. These great thinkers had these misconceptions because they were natural; it took the unusual thinking of the great Newton (and his great predecessors, including Kepler, Galileo, and others) to correct these misconceptions. We should not feel bad if it takes us a few weeks or months to correct our own misconceptions about motion. Newton's laws of Motion 1: If the total of all forces acting on an object is zero, then the acceleration of the object is zero. This means that if the net force acting on an object is zero, if it is at rest it will stay at rest, or if it is moving, then it will continue moving in a straight line at a constant speed. 2: F = ma, which is equivalent to a = F/m; this is a vector equation! This means that if there is a net force acting on an object, then the object will accelerate. The direction of the acceleration is the same as the direction of the net force. The magnitude of the acceleration is equal to the magnitude of the net force divided by the mass of the object. 3: Forces always occur in pairs; even if we only talk about one of the forces in the pair, the partner force is always there. If object A exerts a force on object B, then automatically (and simultaneously) object B exerts a force on object A. The two forces have equal magnitudes and opposite directions. The following examples give you a chance to understand Newton's laws of motion in basic situations, and also to explore and correct some misconceptions that you might have. Consider a hockey puck sliding on very smooth ice; the ice is so smooth that we can assume that there is no resistance whatsoever to the puck's motion. Ch4A Page 2

3 Describe the motion of the puck. Is the motion consistent with Newton's laws of motion? Conclusion: Yes, the puck travels at a constant speed in a straight line, in accord with Newton's first law of motion, until it bumps into something. Consider a marble at rest in a wagon. The wagon moves in a straight line at a constant speed. The wagon suddenly stops. Describe the subsequent motion of the marble. Is this consistent with Newton's laws of motion? Sample student responses: The marble has inertia. The marble continues to move forward once the wagon stops, due to the marble's inertia. An object in motion (in a straight line at a constant speed) tends to stay moving in a straight line at a constant speed unless an external force acts to change its motion. The marble keeps going at a constant speed in a straight line because it has no acceleration. There is no force that pushes a passenger in a car forward when the brakes are applied suddenly. The force on the passenger is actually towards the back of the car; the seatbelt acts towards the back to slow the person down. Consider the following scenario, and its explanation. Is the explanation correct? Forces cause motion. Therefore, if an object is moving at a constant speed, there Ch4A Page 3

4 must be a force acting on it, because the object is moving. Comments: False. It's not quite correct to say that forces cause motion; it's more accurate to say that forces cause changes in motion, according to Newton's second law of motion. If an object is moving at a constant speed, then its acceleration is zero, so the net force acting on it is zero. Sure, some forces might be acting on the object, but the net force is zero. The net force on an object moving at a constant speed is zero, because its acceleration is zero, by Newton's first (or second) law of motion. Consider the following scenario, and its explanation. Is the explanation correct? Forces cause motion. Therefore, if an object is at rest, there must be no forces acting on it. For example, as you sit at rest in your chair, there are no forces acting on you. Comments: False. As we discussed above, it's not quite right to say that "forces cause motion." If an object is at rest, the net force acting on it is zero, but that doesn't mean that no forces are acting, just that the sum of the forces acting on the object is zero. As you sit in your chair, there are at least two forces acting on you, gravity acting downward, and the force that the chair exerts on you upwards. The sum of the two forces is zero; that is, the net force acting on you is zero. Consider the following scenario, and its explanation. Is the explanation correct? Two objects are each moving at a constant speed in a straight line. However, Object A is moving at twice the speed of Object B. Therefore the net force acting on Object A is twice the net force acting on Object B. Comments: False. Because each object moves at a constant speed, the net force on each object is zero, by Newton's first or second law. Of course, the driving force applied to Object A is probably larger than the driving force applied to Object B, but then air resistance on Object A will also be larger than air resistance on Object B. This is what makes it so hard to design cars that go very fast. When you double the speed, the air resistance tends to go up by a factor of 4 (!!), so incremental increases in speed become increasingly difficult. Another reason for the difficulty is that one way to increase speed is to make engines more powerful, but this usual entails more massive engines, and this increases the weight of the Ch4A Page 4

5 car. But road resistance is proportional to the weight of the car, so this also is a great difficulty for an engineer who wishes to design a car that can go very fast. Consider the following scenario, and its explanation. Is the explanation correct? Two objects are each moving in a straight line. The acceleration of Object A is twice the acceleration of Object B. Therefore Object A has twice as much force as Object B. Comments: False. An object cannot "have" a force. If the masses of the two objects are the same, then the net force exerted on A is twice the net force exerted on B, by Newton's second law. Note: Objects can have mass, for example, but objects cannot have forces. A force, on the other hand, is not the property of a single object, like mass, colour, size, etc. A force is an interaction between two objects; one object exerts a force on the other object. Consider the following scenario, and its explanation. Is the explanation correct? A rock tied to a string is swung in a circle. The rock moves around the circle at a constant speed. This is consistent with Newton's first law of motion. The net force acting on the rock is zero; after all, a body in motion tends to stay in motion, according to Newton's first law of motion. Comments: False. The diagram below is meant to convince us that the rock indeed accelerates, even if its speed is constant; after all, its direction continually changes. Remember that velocity is a vector! Thus, the statement is false. Because the rock continuously accelerates, there must be a continuous force acting on it. The diagram suggests that the force acts towards the centre of the circle. Tension in the string is the origin of the force on the rock. This force is an example of a centripetal force, so called because it pulls the rock towards the centre of the circle. Ch4A Page 5

6 Consider the following scenario, and its explanation. Is the explanation correct? When you push gently on a book resting on a desk, the book does not move. When you gradually increase the force you exert on the book, the book eventually begins to move. This is because the force you exert overcomes the force of inertia of the book. Comments: False. The applied force overcomes the static friction force, not the inertia. Inertia is not a force; it's a property of matter, also known as mass. There is no such thing as the "force of inertia;" avoid such phrases, as they don't make sense. Consider the following scenario, and its explanation. Is the explanation correct? When you sit at rest in a chair, the net force acting on you is just the gravitational force that the Earth exerts on you. The chair supports you, but doesn't exert a force on you. (a) True, because chairs can't exert forces; only living things (people and animals) can exert forces. (b) False, because gravity doesn't act on you when you sit in a chair; gravity only acts on you when you jump in the air. (c.) False, because chairs and other inanimate objects really can exert forces on other objects. (d) False, because there are no nets around. Comments: Choice (c.) is correct. Consider the following scenario, and its explanation. Is the explanation correct? An object is thrown upwards. As the object moves upwards, the object slows down gradually because the force of the throw gradually dissipates. Ch4A Page 6

7 (a) True, because if the force of the throw dissipated all at once then the object would stop immediately, and we can see that this doesn't happen. (b) False, because objects only appear to slow down gradually; the decrease in speed actually takes place in regular, abrupt steps. (c.) False, because the force of the throw doesn't dissipate; it's just that the opposing force of gravity takes gradually more effect as time passes, cancelling more and more of the force of the throw as time passes. (d) False, because the force of the throw is not carried along with the object. Comments: Choice (d) is correct. Consider the following scenario, and its explanation. Is the explanation correct? When a truck pulls a trailer, the force exerted by the truck on the trailer is equal and opposite to the force exerted by the trailer on the truck. Therefore the net force is zero, because the sum of the two forces in the previous sentence is zero. Comments: (a) The first sentence is true, by Newton's third law. The second sentence is false, because the forces act on different objects, so adding them makes no sense. (b) The first sentence is false, because if the forces were equal how would they be able to move? The second sentence is also false, because if the net force were zero, how would they be able to move? (c.) The first sentence is false, but the second sentence is true. (d) The first sentence is true, and the second sentence is also true. Comments: Choice (a) is correct. Consider the following scenario, and its explanation. Is the explanation correct? When a hockey stick strikes a puck during a slap shot, the force exerted by the hockey stick on the puck is much greater than the force exerted by puck on the hockey stick. You can see this because the puck goes flying and the stick does not. (a) The first sentence is true, and the second sentence is also true. (b) The first sentence is true, and the second sentence is false. (c.) The first sentence is false. Comments: Choice (c.) is correct. Don't confuse causes and effects. The two forces are equal in magnitude by Newton's third law of motion. The fact that the effects are different are because the masses of the objects are different. Ch4A Page 7

8 Consider the following scenario, and its explanation. Is the explanation correct? After an object is thrown upwards, the net force on the object gradually decreases on the upward journey, is momentarily zero at the peak of the motion, and then gradually increases on the downward journey. Comments: No. The gravitational force is constant. The speed of the object changes, not the force acting on it. Remember, objects don't carry force with them. A correct statement is that the speed of the object gradually decreases on the upward journey, is momentarily zero at the peak of the motion, and then gradually increases on the downward journey. The speed of the object changes gradually; the force acting on it does not. Impulsive forces vs. continuous forces Notice that in some of our examples, applied forces act only for very short times, such as in collisions (the slap shot, for example). These are called impulsive forces. Other forces, such as gravity, act continuously. As you work through the examples to come to a deeper understanding of Newton's laws of motion, asking yourself whether an applied force is impulsive or continuous may help you to understand a given situation. Here are additional questions for your careful study. Deeply understand these situations, and you'll have gone a long way towards a deep understanding of Newton's Laws of Motion. Q1: Two small spherical metal balls of the same size and shape are dropped from the same height, about 10 m from the ground. One of the balls is twice as heavy as the other. Which ball hits the ground first? How much sooner? A lot sooner, or only a little sooner? Why? Sample student responses: The balls hit the ground at the same time, because the velocities of the balls are independent of the balls' masses. Without air resistance the acceleration due to gravity is the same for each ball, so their velocities are the same. Ch4A Page 8

9 the heavier object should hit the ground first under ideal circumstances, they should both hit the ground at the same time if you neglect air resistance, they have the same acceleration (due to gravity), independent of their masses Analysis using Newton's laws of motion: The gravitational force exerted by the Earth on an object of mass m is mg, at least near the Earth's surface. If air resistance is neglected, then this is the only force acting on each of the balls. Thus, by Newton's second law of motion, the acceleration of a ball with mass m is a = F/m a = mg/m a = g Notice that the mass of the ball cancels, and so the acceleration of a freely falling object is the same, no matter what its mass is. (This is what we mean by "free fall:" no other forces act except for gravity.) This is the prediction of Newton's second law, and it has been verified countless times by very precise experiments. If the balls both start from rest at the same height, and both have the same constant acceleration, then they also have the same velocities, and the same position functions. Thus, the two balls have exactly the same motions, and so hit the ground at the same time. Further discussion: Aristotle believed that heavier objects fall faster, because each object has an inherent desire to return to the Earth, and the heavier the object the greater the desire. Galileo argued against this as follows: If we take two identical objects and place them so that they are touching, would they then fall twice as fast, just by virtue of touching? Attach them with a thread if you like, so that they form one single object; do you really believe that the single object would fall faster just because of the thread? And do you really believe that they would suddenly slow down if the thread were cut? Regardless of our beliefs, the final word belongs to experiment: Do the experiment and measure for yourself what happens. Check out the cool video of a cannonball and a feather falling at the same rate in a vacuum chamber: Q2: The same two balls from Q1 are rolled off a horizontal table top with the same initial speed. Which one lands farther from the edge of the table? How much farther? A lot farther or only a little farther? Why? Sample student responses: Ch4A Page 9

10 Because they have the same initial velocity, when they leave the table gravity acts on each ball equally, so the two balls hit the floor the same distance away from the table. The x- and y-components of motion are independent; elaborates on first comment. The effect of friction is the same for both balls, both on the table and in the air. Won't inertia affect the heavier ball more than the lighter ball? The heavier ball goes further because its momentum is affected by its mass; doesn't the heavier ball have greater momentum, and therefore goes further? If you slide two objects on a surface, why does the lighter one goes further? neglecting air resistance, the distances will be the same; gravity acts on the balls independent of their masses maybe we need to account for friction on the table the lighter one travels farther, because there is no acceleration in the x-direction; they will hit the ground at the same time, but the lighter one travels farther in the x- direction Analysis using Newton's laws of motion: As we discussed in the previous question, the two balls have exactly the same acceleration once they leave the table and are falling freely. (Remember that this is only true provided that there is no air resistance; if there is air resistance, the situation is much more complicated.) If they leave the table with the same velocities, at the same time, then their subsequent motions will be identical. Therefore they travel the same distance (and the same time) before they hit the ground. Won't inertia affect the heavier ball more than the lighter ball? Yes. The heavier ball has the greater inertia, but the heavier ball also has the greater force acting on it. The two factors exactly cancel out. (Remember that for a projectile, you can treat the two components of the motion independently.) Note that mass is a measure of inertia, so larger mass means larger inertia. In other words, the larger the mass, the greater the resistance to acceleration. a = F/m a = mg/m a = g The heavier ball goes further because its momentum is affected by its mass; doesn't the heavier ball have greater momentum, and therefore goes further? The heavier ball definitely has greater momentum than the lighter ball, as they have the same velocity. This means that the heavier ball has a greater impact when it hits the ground, so it can leave a larger dent, cause more damage, etc. However, how far each ball goes depends on their velocities, not on their momenta; they have the same velocity at each instant, so they have the same displacements. In Q1 and Q2, note that we must carefully distinguish between causes (i.e., forces) and effects (i.e., accelerations). In Q1 and Q2, the causes are different, because the forces acting on each ball have different magnitudes, but nevertheless the effects are the same because the masses are in the same proportion as the forces. Remember: Ch4A Page 10

11 a = F/m a = mg/m a = g For advanced readers only: Note that there is a logical distinction between two different concepts, that we use the same label m to represent. Inertia, also called inertial mass, is the concept that appears in Newton's second law of motion. Inertial mass represents the property of an object that resists acceleration. Gravitational mass appears in the formula for the weight of an object, mg. Gravitational mass represents the property of an object that is responsible for a gravitational force between two objects. We use the same symbol m to represent these two distinct concepts because all experiments done so far support the hypothesis that these two distinct concepts are numerically equal. Newton was well aware of the logical distinction between inertial and gravitational mass, and did experiments to test their equivalence. Many other experiments have been done over the years to improve the accuracy; here are links to recent tests: Einstein made the equivalence principle (the concept that inertial mass and gravitational mass are equivalent) the foundation stone of his theory of gravity, which supersedes Newton's theory of gravity and includes the latter as a special case. To learn more about early experiments supporting the equivalence between inertial and gravitational mass, try this link: Q3: A heavily-loaded transport truck collides head-on with a very small car. The mass of the truck is much greater than the mass of the car. Which vehicle exerts the greater force on the other vehicle during the collision? A lot greater or only a little greater? Sample student responses: The truck exerts a greater force on the car than the car exerts on the truck. the small car exerts more force than the truck the forces will be dependent on the speeds and the masses by Newton's second law (F = ma), the truck will exert a greater force because it has a greater mass they will exert the same force but the smaller car will have the greater acceleration, because it has the smaller mass Ch4A Page 11

12 maybe they exert the same force on each other, because the forces should be equal and opposite Analysis using Newton's laws of motion: By Newton's third law of motion, the magnitude of the force exerted by the small car on the heavy truck is equal to the magnitude of the force exerted by the heavy truck on the small truck. It doesn't seem this way, because the effects on each participant in the collision are different, and this is what is hard to wrap our minds around. Consider an insect that hits the windshield of a fast-moving car. The force that the windshield exerts on the insect may be enough to destroy it. The force that the insect exerts on the windshield has the same magnitude, but has negligible effect on the windshield. The collision of the heavy truck and the small car is similar. Just because the force of A on B has the same magnitude as the force of B on A, it's not necessarily true that the resulting accelerations of A and B will be the same. The insect's mass is so small that it's acceleration upon collision with the windshield is enormous. The windshield (and the car its attached to) is so massive that it's acceleration after collision with the insect is not noticeable. Similarly, the small car colliding with the heavy truck has a much greater acceleration than the heavy truck. Don't confuse the equal forces on the small car and heavy truck with the unequal effects: the damage done to each, and the acceleration of each. Q4: A ball is swung in a horizontal circle at a constant speed. The string suddenly breaks. As seen from above, which path does the ball follow after the string breaks? Ch4A Page 12

13 Car moving around a circular curve, hits a patch of ice: Analysis using Newton's laws of motion: Once the string breaks there are no longer any forces in the horizontal plane in which the ball was moving. Thus, the ball follows its tendency to continue moving in a straight line at a constant speed, according to Newton's first law of motion. (The ball "flies off on a tangent.") That is, the horizontal component of the ball's velocity will remain constant once the string breaks. (To make the discussion simple, we are ignoring gravity, which acts in the vertical direction.) When you are using an electric mixer to mix the batter for a cake, if you bring the mixer out of the bowl, the batter will splatter from the mixer in all directions. But observe this carefully and you will see that the batter does not splatter radially, but more in a pinwheel pattern: Ch4A Page 13

14 The splatter from the mixer can also be explained using Newton's first law. Just like the ball on a string whirling in a circle, or the car driving around the curve, the batter is held to the mixer's blades by attractive forces (it's "sticky"). However, if the sticky forces are not strong enough, then the batter releases from the blades, and "flies off on a tangent." Industrial centrifuges work similarly; the lettuce spinner you have in your kitchen works in the same way. Q5: A hockey puck is sliding with constant speed in a straight line from A to B on a frictionless horizontal ice rink. When the puck reaches B, it receives a brief hit from a hockey stick in the direction of the arrow. Here's a view from above: Which path does the puck follow after being struck by the hockey stick? Explain. Sample student responses: If the applied force from the stick is greater than the previous force, then the puck will go in the direction of the greater force. (No. To apply Newton's second law, you substitute the vector sum of all forces acting on the object for F in the formula. But Ch4A Page 14

15 once the puck is moving towards the right, the first force is gone; the first force is not carried along with the puck, and no longer acts on the puck. When the stick hits the puck later, it is the net force acting on the puck, and it only acts for a short time.) If the two forces are perpendicular, then wouldn't the result be somewhere in between? (Remember, the two forces do not act at the same time. To analyze the subsequent motion of the puck after the second force acts, you only need to use the second force.) Wouldn't you treat this using vectors? (Yes.) The change in direction and force would be the "net" of the initial force and the applied force from the stick. (No. When the second force acts, it is the net force acting. The first force no longer acts.) Wouldn't Newton's third law be applicable here? There is a countering of forces; when the stick hits the puck, the puck also acts on the stick, and when we look at the direction that the puck is going in, we're looking at acceleration not force. (No. Consider the force that the puck exerts on the stick and the force that the stick exerts on the puck. These two forces act on different objects, and so we don't add them up. Only add forces acting on one object.) Isn't it possible to make the puck go North if you hit the puck very hard? (Yes, but you can't hit the puck in the direction shown. You must hit the puck in a different direction, as shown in the figures below: Analysis using Newton's laws of motion: Initially the puck has an x-component of velocity, but zero y-component of velocity. The force exerted by the stick on the puck is entirely in the y-direction; there is no x-component of force. By Newton's second law of motion, the puck will experience a brief acceleration in the direction of the applied force (that is, the y-direction) while it is in contact with the stick. There is no acceleration in the x-direction. Thus, the x-component of the puck's velocity remains constant, and the y- component of the puck's velocity increases while the puck is in contact with the stick. Once the puck leaves contact with the stick, the net force acting on it is zero (gravity and the normal force exerted by the ice on the puck balance to zero). Thus, after the puck leaves contact with the stick, its velocity remains constant, and so it continues to move in a straight line at a constant speed. Thus, "b" is the correct path. Ch4A Page 15

16 Q6: When the puck is moving on the frictionless path you have chosen in the previous question, the speed of the puck 1. is constant. 2. gradually increases. 3. gradually decreases. 4. increases for a while and then decreases. 5. is constant for a while and then decreases. Explain. Analysis using Newton's laws of motion: Some students believe that the speed of the puck will decrease because the force carrying the puck along in its motion dissipates. There is no such force; once the puck leaves the stick, the net force is zero. Thus, the speed of the puck is constant after it leaves the stick, in accord with Newton's first law of motion. Remember that objects do NOT carry forces along with them. Forces act on objects, but objects don't have forces. An object can have mass, have a colour, have a shape, have a position, and so on, but an object cannot have a force. As usual, one of the difficulties in learning physics is to understand the precise way language is used in physics, which is sometimes different from the every-day usage of language. In every-day language, we might say an argument is forceful, or carries a lot of force, or that a person is a force of nature, but this is not the way we use the word "force" in physics. Q7. A ball is thrown horizontally from the top of a cliff. Which path does the ball take? Explain. Ch4A Page 16

17 Analysis using Newton's laws of motion: Path number 2 is correct. The horizontal component of velocity is constant, but the speed in the vertical direction steadily increases. Thus, the path of the object is not a straight line, and so this eliminates path 1. Paths 3, 4, and 5 can be eliminated because they suffer from the "Coyote and Roadrunner" problem; that is, the object goes straight out from the cliff for a while. This could only happen if there were no net force acting on the object; we know this is incorrect, because gravity acts on the object, so it begins to accelerate immediately. Thus path 2 is correct. Q8: An engine accidently falls off an airplane as it is in flight. Which is the path of the engine as it falls to Earth (from the perspective of an outside observer)? Explain. Sample student responses: 4 is the most popular choice. Analysis using Newton's laws of motion: This is virtually the same situation as in the previous problem. When the engine leaves the airplane, its initial velocity is the same as the airplane's velocity. Thus, the engine behaves in the same way as the ball thrown from the top of a cliff in the previous problem, for the same reasons. Q9: You are below-deck on a ship, or in an airplane. The ship or airplane moves at a constant speed in a straight line. You reach overhead and drop a ball. The ball lands behind you? in front of you? at your feet? Depends on the speed of the ship/airplane? Depends on something else? Explain. Ch4A Page 17

18 Analysis using Newton's laws of motion: You will certainly have experienced being in a car moving at an approximately constant velocity on a highway, or being on an airplane moving at an approximately constant velocity. (And by "constant velocity" we mean "constant speed moving in a straight line".) If you toss an object inside the car or inside the airplane, it behaves as if you were at rest in your living room at home. In other words, being in a car or airplane moving at a constant velocity, objects inside the car behave just as they do "at rest" in your living room. Another way to say this is that the car and the airplane are examples of inertial reference frames. An inertial reference frame is one in which Newton's first law is valid. Once you find one such reference frame, any other reference frame moving with constant velocity relative to it is also an inertial reference frame. Thus, the ball drops straight down, hitting the ground at your feet. Q10: A large truck breaks down on the road and receives a push back to the station by a small car. Compare the force that the car exerts on the truck with the force that the truck exerts on the car when the car is accelerating up to its cruising speed. Analysis using Newton's laws of motion: By Newton's third law of motion, the magnitude of the force exerted by the car on the truck is the same as the magnitude of the force exerted by the truck on the car. Q 10A: But don't these two forces cancel, so that the net force is zero? How then could the vehicles accelerate? No, the forces do not cancel, because they act on different object. Remember that all forces occur in pairs, acting on different objects. If A exerts a force on B, then B exerts an equal and oppositely-directed force on A. One force acts on A, the other force acts on B, so the two forces do not cancel. Q11: A large truck breaks down on the road and receives a push back to the station by a small car. Compare the force that the car exerts on the truck with the force that the truck exerts on the car when the car is moving at a constant cruising speed. Analysis using Newton's laws of motion: By Newton's third law of motion, the magnitude of the force exerted by the car on the truck is the same as the magnitude of the force exerted by the truck on the car. Q12: An elevator is being lifted up at a constant speed by a cable. Compare the Ch4A Page 18

19 force exerted by the cable on the elevator to the gravitational force exerted by the Earth on the elevator. (All other forces are negligible.) Explain. Analysis using Newton's laws of motion: The only forces acting on the elevator are gravity and the force exerted by the cable on the elevator. Because the elevator is moving at a constant speed in a straight line, the net force acting on the elevator is zero, according to Newton's first law. Thus, the gravitational force on the elevator and the force exerted by the cable on the elevator add up to zero; thus, the two forces have the same magnitude and opposite directions. Note the difference between this question and Q10A. In Q10A, the two forces being considered act on different objects, so we are not allowed to add them together. In Q12, the two forces being considered act on the same object, so we are allowed to add them to determine the total force acting on the object. Q13: Despite a very strong wind, a tennis player hits a tennis ball with her racquet so that the ball passes over the net and lands in her opponent's court. After the ball leaves the racquet, and before it hits the ground, which forces act on the ball? Gravity? The force from being struck by the racquet? A force exerted by the air? Other forces? All of the above? None of the above? Explain. Analysis using Newton's laws of motion: Once the ball leaves the racquet, and before it hits the ground, the only forces acting on the ball are the gravitational force exerted by the Earth on the ball, and the force from the wind. There is no force that "follows" the ball in its motion; once it leaves the racquet, there is no more force exerted by the racquet on the ball. Remember, forces are not carried by objects; forces are exerted on objects. OK, now that we have looked at a number of specific examples, let's dig into the key concepts of Newtonian dynamics. Force What is a force? One definition of a force is a physical influence of one object acting on another object. An object can "have" energy, an object can "have" momentum, an object can "have" mass or velocity, but an object CANNOT "have" force in the same way. Remember we are speaking about the technical, physics definitions of these words, not the every-day meanings of these words. Ch4A Page 19

20 An object can exert a force on another object, but an object cannot have force. A force always acts between two objects; one object exerts a force on another object, and the second object "feels" the force exerted by the first object on it. We often think of forces as pushes or pulls. A force requires an agent; that is, something must be doing the pushing or pulling. Forces can be modelled as vectors; that is, a force has magnitude and direction. This is confirmed by numerous careful experiments. The SI unit of force is the newton (N). A short catalogue of forces (see the textbook): weight; i.e. the force that Earth's gravity exerts on an object The magnitude of the gravitational force that the Earth exerts on an object of mass m can be written as W = mg or, equivalently, as F = mg; this is an excellent approximation provided that the object is not too far from the Earth's surface. As was introduced in the previous chapters, g is the acceleration due to gravity, and has a magnitude of approximately 9.8 m/s 2. CONVENTION: When we write g, we mean only the magnitude of the acceleration due to gravity, without direction, and therefore without any positive or negative sign. spring force If you pull on a spring, the spring pulls back on you. If you push on a spring, it pushes back on you. The magnitude of the force exerted by the spring is proportional to the amount by which it has been stretched or compressed, and in the opposite direction of the stretch or compression: F = kx This relationship (known as Hooke's law) is an excellent approximation provided that the magnitude x of the stretch or compression is not too large. The constant of proportionality k is called the stiffness of the spring, and is larger for stiffer springs. Physics majors: It turns out that many situations can be modelled by forces that satisfy the same relationship as Hooke's law, even though no springs are involved. For example, one can model inter-atomic forces using a similar relation, with good results; that is, just pretend that the electrical forces acting between atoms are just springs, use the same mathematics as for springs, and sure enough your predictions about how the atoms behave is pretty good. This happens over and over again in physics; the same mathematical models come up over and over again in different situations. The model of small-amplitude vibrations experienced by a small mass attached to the end of a spring, called simple harmonic motion, is a classic paradigm in physics. We'll explore this model in Chapter 14, later in the course. The same mathematics is Ch4A Page 20

21 also used to describe everything from electrical oscillations in tuned circuits, reception and transmission by antennas, and many other kinds of oscillations. There are even quantum versions of simple harmonic motion that are applied to microscopic systems, such as the vibrations of atoms and molecules. Even the fundamental behaviour of electromagnetic fields in quantum field theory uses the quantal version of simple harmonic motion as a basic model. The moral: Learn the simple harmonic motion paradigm well, because it is used all over physics. tension force When a string is pulled from each end, or is attached to a fixed object and is pulled from the other end, we say the string is under tension. If you hang a chandelier from the ceiling, the string or chain that holds the chandelier up exerts an upward force on the chandelier. Such forces are called tension forces. normal force When you sit on a chair the Earth exerts a gravitational force on you downwards. You would fall through the chair unless the chair exerted an upward force on you to balance the gravitational force. The force exerted on you by the chair is called a normal force, because it acts perpendicular to the surface of the chair; "normal" is another word for "perpendicular". friction Each surface, no matter how smooth, as microscopic protrusions. When two surfaces rub against each other, the protrusions from each surface bump into each other; this manifests macroscopically as a resistive force that we call friction. Can you imagine a world without friction? That might seem great, but friction has many benefits in our world, some of which we might take for granted. For example: If there were no friction, would any knot stay tied? How would you get moving if you were standing on a frictionless surface? The lock-washer: The purpose of the lock washer is to force the threads of the nut against the threads of the bolt, so that the normal force between them is increased; this increases the frictional force between the nut and the bolt, making it harder for them to loosen. (As we'll see next week, the frictional force between two objects pressed together is proportional to the normal force.) In effect, a lock washer is a kind of spring, as you can see in the following image: Ch4A Page 21

22 drag Drag is another word for "air resistance", except that drag is more general, because it can apply to other fluids too. For example, when you feel resistance while walking through water in a swimming pool, this is a type of drag. engine thrust electric forces magnetic forces Classification of force types contact forces and non-contact forces Forces such as normal forces, tension forces, spring forces, friction and drag, and similar forces, are called contact forces because they arise because two objects are touching. However, electrical forces, magnetic forces, and gravitational forces are more mysterious, because they operate even though the object exerting the force and the object on which the force is exerted are not in contact. four fundamental types of forces: gravitational force, strong nuclear force, weak nuclear force, electromagnetic force The two "nuclear" forces are "short-range" forces that act inside atoms; we won't discuss them any further in this course. We call them "short-range" forces because they diminish in magnitude so rapidly with distance that for all practical purposes we can pretend that they don't exist outside of atoms. The other two types of forces, gravitational and electromagnetic, are "long-range" forces. In other words, although their magnitudes diminish with distance, the decrease in their strength is not so rapid, so that their effects can still be felt over very long distances. Ch4A Page 22

23 Virtually all of the forces that we deal with in this course that are not gravitational are ultimately electromagnetic. For example, the forces that hold atoms together in molecules (and that hold molecules together) are electromagnetic. Thus, you can pretend that your chair is somewhat like the top of a trampoline, where the atoms are connected by minute springs. When you sit in the chair, it's as if the surface of the chair flexes a little, and exerts an elastic force upward on you. In reality, there are no springs; what holds the atoms together is electromagnetic forces. Thus, normal forces are ultimately electromagnetic in origin. Similarly, friction, tension, and spring forces are all electromagnetic in origin. the theme of unification in physics In modern physics, a theme that has resulted in a lot of progress in our understanding of the universe is that of unification. For example, in the 1860s, Maxwell unified (to some extent) electricity and magnetism, which were before then thought to be two separate interactions. Thanks to his work, in which he (among other things) predicted that light is an electromagnetic wave, a sort of dance involving oscillations in both electric and magnetic fields, electricity and magnetism were recognized as being intimately connected. This intimate union was recognized as being even closer thanks to the work of Einstein (with his theory of special relativity in 1905), where the difference between electric and magnetic fields was seen to be simply one of perspective. Einstein then embarked on the task of unifying electromagnetic fields with gravitational fields; that is, he hoped to show that each type of field was simply an aspect of some deeper kind of field, which could appear to be electromagnetic from some perspectives and could appear to be gravitational from other perspectives. Alas, Einstein was never able to unify electromagnetic fields with gravitational fields. To make matters more complicated, the two nuclear force fields were discovered late in his life. Interestingly, the two nuclear force fields were unified with electromagnetic fields thanks to the work of many physicists (crowned by the work of Weinberg, Glashow, and Salam in the 1970s). Gravity remains the oddball. A related problem is to develop quantum versions of all physics theories about fundamental force fields. We have excellent quantum theories for electromagnetic fields, and for the strong and weak nuclear force fields, but as yet there is no quantum field theory of gravitational fields. (String theory is an attempt, but it is not yet a theory, and so remains only a tantalizing search in the dark.) Newton's first law of motion Modern statement: If the total force (net force) acting on an object is zero, then (a) if the object started out at rest, it will remain at rest, and (b) if the object was moving in a straight line at a constant speed (i.e., if its velocity was constant) then the object will remain moving in a straight line at a constant speed. Ch4A Page 23

24 In Newton's own words: Every body perseveres in its state of being at rest or of moving uniformly straight forward except insofar as it is compelled to change its state by forces impressed. (see < Newton's first law of motion is also called the law of inertia. Inertia is the tendency of an object to remain at rest or moving in a straight line at a constant speed. That is, if an object is not acted upon by external forces, then it will remain in a state of inertia, which means it will continue in a straight line at constant speed (if that was its initial state), or it will remain at rest (if that was its initial state). Physics majors: Note that there is more to Newton's first law of physics than we have described so far. Newton's first law is not just a special case of Newton's second law; that is, if we substitute 0 for F in Newton's second law, we obtain a = 0, which is equivalent to Newton's first law: If F = 0, then a = 0. Newton's first law is more than this. Newton's first law asserts the existence of a reference frame in which it is valid. That is, there exists a reference frame in which if the net force on an object is 0, then its acceleration is also 0. Such a reference frame is called an inertial reference frame. Furthermore, any reference frame that moves with constant velocity relative to an inertial reference frame is also an inertial reference frame. Thus, as soon as we find one inertial reference frame, there are in infinite number of others. In practice, we typically work with reference frames that are only approximately inertial. For example, the reference frame that is at rest with respect to you while you are sitting at a desk in your home is approximately inertial, but not exactly so, because the Earth rotates; rotation is a kind of acceleration, and fouls things up. Imagine being on a carousel; strange things happen, even though there are no forces acting. The carousel is not an inertial reference frame. You'll learn how to deal with non-inertial reference frames in second-year mechanics. There are important consequences to being in a rotating reference frame; for example, the typical cyclone action of winds in the northern hemisphere is due to the rotation of the Earth. Do a search on the term "Coriolis forces" if you would like to learn more about this, but remember that Coriolis forces are not real forces, but only apparent forces, and arise because we observe phenomena from the perspective of a non-inertial reference frame. Similarly, when you feel yourself thrown forward in a braking car, or thrown towards the passenger door in a car that is turning, you are feeling fictitious forces because your perspective is a non-inertial reference frame. The forces are not real, we just think they are because of our perspective. Examples of Newton's first law: a book resting on a desk; gravity acts downward and the normal force from the desk acts upward; the net force on the book is zero, so the book remains in a state of rest shovelling snow or gravel: you accelerate the gravel, but then stop the shovel abruptly; the Ch4A Page 24