Ch4 Page 1 Chapter 4: Forces and Newton's Laws of Motion Tuesday, September 17, 2013 10:00 PM In the first three chapters of this course the emphasis is on kinematics, the mathematical description of motion. In Chapters 4 and 5, the heart of the course, we shift the emphasis to explaining why changes in motions occur. This is called dynamics. Together, what we have learned and what we will learn, kinematics and dynamics, form the core 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 nearly two millennia of his followers, 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 connected 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 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
Ch4 Page 2 philosophy involved careful reading of Aristotle and commentaries and discussions of his works. 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 comments, 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. 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? 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?
Ch4 Page 3 Q3: A heavily-loaded transport truck collides head-on with a very small car. The mass if 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? 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?
Ch4 Page 4 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.
Ch4 Page 5 a. b. c. d. e. Q6: When the puck is moving on the frictionless path you have chosen in the previous question, the speed of the puck is constant. gradually increases. gradually decreases. increases for a while and then decreases. is constant for a while and then decreases. Explain.
Ch4 Page 6 Q7. A ball is thrown horizontally from the top of a cliff. Which path does the ball take? Explain. 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? Explain.
Ch4 Page 7 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. 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.
Ch4 Page 8 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. Q12: An elevator is being lifted up at a constant speed by a cable. Compare the force exerted by the cable on the elevator to the force exerted by gravity on the elevator. (All other forces are negligible.) Explain
Ch4 Page 9 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. Force What is a force? 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 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 the 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
Ch4 Page 10 A short catalogue of forces (Section 4.3 in the textbook): weight spring force tension force normal force friction drag engine thrust electric forces magnetic forces classification of force types contact forces and non-contact forces four fundamental types of forces the theme of unification
Ch4 Page 11 Newton's first law of motion: reference frames Examples: contrast "getting into motion" vs. "remaining in motion"; getting into motion from rest requires a force; remaining in motion (as long as your going in a straight line at a constant speed) does not require a force only external forces can move your centre of mass; external forces change the motion of an object; internal forces are not effective at changing the state of motion of your centre of mass Example: sailboat pushed by electric fan (contrast "Everglade" boat, and rocket propulsion) Newton's second law of motion: What can forces do to an object? They can cause an object to: speed up slow down change its direction of motion All three of these possible types of effects can be categorized as accelerations. Thus, acceleration means more than just speeding up or slowing down. Newton's second law of motion has several aspects. The first is that the resulting
Ch4 Page 12 acceleration is in the same direction as the net force. The second is that the magnitude of the acceleration is related to the magnitude of the force. calculus lovers: the mechanical view of the world via Newton's second law Newton's third law of motion: forces always come in pairs Examples: push a skater in the back, paddle a canoe Free-body diagrams a tool for analyzing problems involving forces and Newton's laws of motion we'll look at them in more depth in Chapter 5 Some further random thoughts: "Motion is caused by forces." pretty vague "Acceleration is caused by forces." this is more precise, and it is also correct "Motion in a straight line at a constant speed happens for no reasons." to contrast with "everything happens for a reason"; well, this statement is also a bit vague, but it's OK if you interpret it correctly; it gets back to what is "natural";
Ch4 Page 13 according to Newton's first law, an object at rest or moving at a constant speed in a straight line will keep doing what it's doing unless a force acts in other words, there is no need to explain motion in a straight line at a constant speed; it occurs if there are no forces acting Some final comments on the mechanical world view according to Newton: determinism vs. free will "We ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence that could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it an intelligence sufficiently vast to submit these data to analysis it would embrace in the same formula the movements of the greatest bodies of the universe, and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes." Laplace, 1814 this deterministic mechanical world view was already shown to be impossible in principle by Einstein's special theory of relativity (1905), and was subsequently strongly contradicted (for a quite different reason) by quantum mechanics (1920s); and let's not forget chaos theory, developed starting in the late 20th century, where small changes in initial conditions lead to widely divergent final conditions; this means that measuring the initial positions and velocities of all the particles in a system (a la Laplace) is insufficient for determining all later positions, even approximately; that is, inevitable small measurement uncertainties in initial positions and velocities make the final positions and velocities unpredictable Interestingly, Newton was a dualist. He believed that the universe contained two types of substances, matter (which is subject to the laws of physics) and mental substance, such as the soul (which is subject to free will). Currently, the fundamental laws of microscopic physics are non-deterministic, and the question of free will is still open, although it seems that we have free will. It's also not clear whether science can shed light on the issue, but one never knows what will be discovered by one of us inventive humans down the road.
Ch4 Page 14 the mechanistic program according to Newtonian mechanics (for calculus lovers only): think of a = F/m as a differential equation fits in with determinism