PHYSICS 107. Lecture 5 Newton s Laws of Motion

Transcription

1 PHYSICS 107 Lecture 5 Newton s Laws of Motion First Law We saw that the type of motion which was most difficult for Aristotle to explain was horizontal motion of nonliving objects, particularly after they've ceased contact with the thing that put them in motion in the first place. Aristotle resorted to a complicated mechanism in which the medium itself somehow maintains the motion. This was the only way that he could maintain two of his most important principles: the fundamental distinction between living and nonliving things, and the idea that the natural state of motion for nonliving objects is up or down, then remaining at rest. Newtonian physics denies both of these principles. However for the moment we will only talk about Newton s disagreement with the second principle. Newton explicitly denied that the only natural state of motion is rest. His idea was that the natural state of motion is either rest or motion at constant speed, what we might think of as coasting. The statement of Newton's first law (translated from the original Latin) is: Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed. We can state this more simply by just saying that s, the speed of an object, is equal to a constant unless the object is acted upon by something else. When I say that something is equal to a constant what I mean is that it does not change with time. So constant speed means that the object is not speeding up or slowing down, it's on cruise control. Notice that the state of rest is just a special case of constant speed: the speed just happens to be zero.

2 To reach this new conception of the natural state of motion as being constant speed, we need to think of friction in a new way. Friction must now be regarded as being itself a kind of force. Indeed, I think this is quite natural. Surely when a book slides across the surface of a table, there is a "force impressed" by the surface on the book. If the book comes to rest, that's not because rest is a natural state. Rather it's the effect of the surface on the book. Different surfaces have different forces of friction with correspondingly different motions for objects sliding across them. But in Aristotle s view, friction was somehow not a separate influence on a moving object. Rather, it was just a part of the object s coming to its natural state of rest. Still, we mustn't be too hard on old Aristotle. Except for celestial objects (which he did not even think of in the same category), he did not really have the opportunity to observe objects that were acted on neither by friction, so he never really observed motion at constant speed. Thus it is not surprising that he never took the step of thinking about motion at constant speed as being natural. But at this point we're going to stop apologizing for Aristotle and bid him goodbye. Newton now tears up his books. Speed and Velocity The quick statement of Newton's first law is that s is a constant unless a force acts. If something does act on the body then s changes. It increases if we push, or decreases if we pull or if friction acts. So now we need a way of describing changes in speed and that brings us to a discussion of velocity and acceleration. Let's imagine a yardstick with all the inches marked off, 0 to 36. Now move your finger from the left to the right on the yardstick starting at 1 and going to 4 and doing this in a second. Then clearly s is equal to 3 in/s. Now start at 10 and go to 6, again taking one second. Then s is equal to 4 in/s. However by using the slightly richer concept of velocity instead of speed we can give a more precise description. We define v = (x final x initial )/(t final t initial ), or v = Δx/Δt for short.

3 Then for our first example we find that v is equal to 3 in/s but for the second example we find that v equals -4 in/s. Motion to the right has positive velocity but motion to the left has negative velocity. This distinguishes the velocity v from the speed s. s is always positive for this kind of motion. In fact we have that s is equal to the absolute value of v, which is written as s = v From now on I am now going to switch to using v instead of s, since it gives a fuller description of the motion. Another tool we can now use to describe motion is that of a graph. Let x be the position. We plot x(t), which is said in words as the function x of t, on the vertical axis and t itself on the horizontal axis. (These graphs and the idea of describing motion as a mathematical function are not due to Newton, but rather to various medieval commentators on Aristotle. These were people who were trying to explain and make explicit Aristotle's theories, and translate them into mathematics.) So the graph of x(t) for an object moving at constant velocity is a straight line and the slope is just the velocity. The slope is positive if the motion is to the right and the slope is negative if the motion is to the left. We can also describe changes in speed in terms of graphs but I will leave that for the discussion sections. Acceleration We can now go further and quantify the concept of changes in speed. Any change in speed is defined to be acceleration. So any time the speed is not equal to a constant we know two things: Newton s first law tells us that something is acting on the body, exerting some kind of a force on it, and we know that there is acceleration, by definition. So let's say that v does change with time. What's the simplest way that this could happen? Well how about v is equal to a times t: v=at? This means that the speed increases proportionally to the time. If a is very big, then the object is really speeding up quickly. If a is small then it is speeding up, but not quite so fast. Again there is a special case. When a=0 then s is equal to constant and that s what happens when there is no force. When a is negative then the final velocity is less than the initial velocity, so that's deceleration. In other words deceleration is negative acceleration. We can now write a = Δv/Δt.

4 Here's an example of constant acceleration. A typical criterion for how fast a car is is the time that it takes to go from 0 miles an hour to 60 miles an hour. A really fast car can do this in about 4 seconds, but a normal car off the lot of your local Ford dealership takes about 8 seconds. What's the acceleration? First of all let's convert 60 miles an hour to feet per second: we find that 60 miles an hour is equal to 88 ft/s, so the acceleration equal to Δv/Δt = (v final v initial )/(t final t initial ) = (88 0)(8-0) ft/s 2 = 11 ft/s 2. Here s another example: if I drop the ball it falls at g = 9.8 m/s 2 which is 32.2 ft/s 2. The acceleration due to gravity is greater than the acceleration of the car by about a factor of three. Vectors and Components We went from the notion of speed to the notion of velocity because velocity can also give a mathematical description of backward motion. But even then we can only describe motion along a line. To talk about motion in real three-dimensional space we need the idea of a vector. Here is an example. Throw a ball straight up in the air and observe its trajectory. Clearly it's not going at a constant speed. It climbs (positive velocity), reaches a certain height, turns around and goes down (negative velocity) until it hits the ground. It's acted on by the force of gravity, so now that we have absorbed Newton s First Law, it's no surprise that the motion is not at constant velocity. However, one number x(t) is enough to know where it is at any time. But if I throw it up and towards the east it goes up and down but at the same time it goes continually towards the east: the motion appears to be a combination of an eastward motion at constant speed and a vertical motion that is not constant at all. How do we describe this? The answer is: resolve the velocity and the position into components. We have an east-west component, a north-south component, and a vertical component. We write the position as r(t) = (x(t), y(t), z(t)), and there is a corresponding velocity

5 v(t) = (v x (t), v y (t), v z (t). v is the change in position per unit time. More specifically, v x is the change in x per unit time, v y is the change in y per unit time, and v z is the change in z per unit time. While we re at it, we may as well do the same for the acceleration: a(t) = (a x (t), a y (t), a z (t). a x is the change in v x per unit time, a y is the change in v y per unit time, and a z is the change in v z per unit time. Vectors are quantities which are resolved into three different directions. I ll write them in boldface in these notes. Motion along a line does not require vectors. Motion in a plane can always be described by two, rather than three numbers, so we might speak of a two-dimensional vector. But usually a vector, for us, will have three components.