Physics 380: Physics and Society Lecture 2: Newton s Laws, Mass, Force, and Motion

Transcription

1 Physics 380: Physics and Society Lecture 2: Newton s Laws, Mass, Force, and Motion [Instructor] Slide #1 Physics and Society Physics 380: Physics and Society Lecture 2: Newton s Laws, Mass, Force, and Motion In lecture two of Physics and Society, we re going to be talking about Newton s Laws, mass, force, and motion. Slide #2 Topics for This Lecture Topics for This Lecture Mass Force Motion: acceleration, speed, and displacement Newton s three Laws Equations of motion One-, two-, and three-dimensional motion The monkey and the gun Complexity versus approximation Friction In this lecture and the next few lectures, we ll be talking about some of the introductory material about physics and introduce some of the basic laws. In this lecture, we re going to discuss mass, force, motion (which includes acceleration, speed, and displacement), we re going to talk about Newton s three Laws, equations of motion, one and two-dimensional motion, we re going to demonstrate an experiment called the monkey and the gun, we re going to talk about complexity versus approximation, and we re going to talk a little bit about friction. Slide #3 Mass Mass is the measure of the quantity of matter an object contains. By definition, a mole of 12 C atoms (6.023x10 23 atoms) has a mass of 12 grams. Inertial mass versus gravitational mass they are the same to high precision. 1

2 Let s talk about the quantity called mass. Mass is the measure of the quantity of matter an object contains. We can differentiate mass from weight. Weight is the effect that gravity has on a mass. An astronaut floating in space appears to have no weight, but, in fact, he has the same mass whether he s in space or on the surface of the earth. We ll see the equations that produce weight in a little while. By definition, the mass scale is derived from a certain number of carbon atoms. That number is called Avogadro s number and 6.023x If we take Avogadro s number of atoms of carbon 12 that is by definition 12 grams of matter. All other masses are derived from that scale. Now a little bit later, we re going to introduce the concept of inertial mass and gravitational mass and I just want to say at this point that we ll see that this quantity called inertial mass shows up in equations dealing with motion is the same mass, nearly as we can determine experimentally, as gravitational mass. That is the mass that produces weight due to the presence of gravity. We ll see these equations in a little while. Slide #4 Force Force: an external influence that causes an acceleration a push or a pull. Forces are vectors and add as vectors F=F 1 +F 2 [Image] Diagram of vectors The next concept we want to introduce is that of force. A force is an external influence that causes the acceleration of an object. We can just say it s a push or a pull. Forces are vectors. That means that they have a magnitude and a direction so we can push an object with a certain force to the left or to the right or to the south or east; it has a direction as well as a magnitude. Forces can add together like vectors. In the first example shown here, we have a force, F 1 and F 2, each of which are acting on a mass pulling it in opposite directions. I ve indicated that force one is bigger than force two by the length of the vector that s showing the arrow and that would produce, in this example, an acceleration of that mass to the left. Acceleration means that the object will start from where it is at rest and move faster and faster to the left. Now if I turn those two forces around and make F 1 push to the right and F 2 to the left, F 1 still being the larger force then the acceleration will reverse also. So forces add, they add like vectors. That is, we have to take into account their direction as well as their size. Slide #5 Force in Nature There are only 4 basic forces: o Gravitational force o Electromagnetic force o Strong force o Weak force Now when we talk about forces, it turns out that there are actually only four basic forces in 2

3 nature. They are the gravitational force, the electromagnetic force, the strong force, and the weak force. Let s start with the gravitational force. You re all familiar with gravity; you re held down to the earth. If you pick up an object and drop it, it accelerates toward the center of the earth. By accelerate, you know that the object starts out slow and gets faster and faster as it moves down. The gravitational force acts between any two masses. The mass of the earth provides the gravity that we observe and an object that has a certain mass gets pulled by that force. That s the gravitational force. The electromagnetic force is responsible for almost all the other phenomena we are familiar with when we talk about force. Electromagnetic means that anything to do with electricity and magnetism, for example, are part of this force. So when you produce static electricity and notice that two things are attracted to each other, that s an example of the electromagnetic force. It s the electromagnetic force that holds atoms and molecules together. The electrons are attracted to the protons in a nucleus and that is the electromagnetic force. The electromagnetic interaction between electrons and different atoms can attract two atoms together forming a molecule. Again, that s the electromagnetic force. So anything dealing with chemistry is the electromagnetic force. In fact, anything dealing with mechanics is the electromagnetic force. That is, if I pull on a rope to move an object, that rope is held together by the forces between atoms within the rope which is held together by the electromagnetic force. So as I say, most of the things you experience in your day to day life are due to the electromagnetic force. You re also familiar with magnetism that causes iron, for example, to be attracted to a magnet. That s another example of the electromagnetic force. The third force, the strong one, it s called strong because it s actually the strongest of the forces, but only on a microscopic scale. The strong force is what holds the nucleus of an atom together. It s the force that attracts protons and neutrons to each other within a nucleus and keeps that nucleus together. It s the energy of the strong force that produces radioactive decay. It s the energy within the strong force that produces the atomic bomb. It s that same strong force that produces nuclear power from a nuclear power plant; that is the strong force. Now the weak force, by name, is weaker than the others, but again only at the microscopic scale is more obscure. It is not something you will run across in your day to day activities, but the weak force is actually very, very important to life on earth because the weak force is responsible for the process called beta decay and it s also responsible for the phenomenon by which stars generate energy. Our sun fuses hydrogen into helium to make energy; that s what makes the sun shine. That process by which two protons fuse is caused by the weak force. Thus, the weak force, even though it s obscure, if very important for our day to day lives. In general, we will not speaking much about these four forces, but I just wanted you to understand that beneath the laws of physics these are the four things that produce force; the four types of forces that produce all phenomena with which we are familiar. Slide #6 Motion Force produces an acceleration (a) Acceleration changes the velocity (v) Velocity over time produces a displacement in position (x, y, z) [Image] Two diagrams/examples of acceleration 3

4 Now let s touch on the subject of motion. Forces, a push or a pull, produces an acceleration of an object. The acceleration produces a change in the velocity of an object. Remember that velocity is a combination of the speed, which is a number, and a direction. So force produces an acceleration; acceleration causes a change in the velocity of an object. If we have an object that s moving with a velocity, after a certain time period it has undergone a displacement in space; it s changed its position in x, y, and z, the coordinates of space. So in one dimension we can draw a plot shown here of time on the horizontal axis and the position x on the vertical axis and an object can move through position and time. It can start off moving in the positive x direction and then it can stay in the same location for a little while and then it can move off into the positive x direction again. You re all familiar with motions. In order for those changes in position with time to occur, that object had to undergo changes in its velocity and thereby it underwent acceleration at various points. Now on the right of this example here, I have two vectors representing velocities, velocity one and velocity two. In the top pair, at time one velocity one has a certain length representing the size of that velocity and at time two, a little bit of time later, velocity two for that object is bigger, longer than velocity one. That velocity has undergone an acceleration to the right so that its velocity increased from time one to time two. It s still going in the same direction, but the magnitude of the velocity changed. The second example is the two vectors underneath. At time one velocity one is the same as it was above, but the second velocity, V 2, in this case, is going in a different direction with the same size, the same length as that vector. So that s another example of acceleration. Acceleration doesn t have to change the size of the velocity; it can change its direction. So acceleration produces a change in velocity either in magnitude or direction. Slide #7 Motion in One Dimension Use to represent change in Acceleration changes the velocity: v Velocity changes position: x o a= v/ t o v= x/ t A car moves in a constant speed in a constant direction: a= v/ t=0 A car moves 10m in 5s: average velocity: v= x/ t=10m/5s=2m/s Let s talk a little bit more about motion in one dimension, but I wanted to mention in the previous example where we had an x direction only. We use the symbol delta, capital Greek delta, to represent change in so acceleration changes the velocity and we symbolize that by v. Delta v is the change in velocity. When velocity is present, we get a change in position, delta x. That is, if v, velocity, is not zero, the position in x changes. We can represent acceleration and velocity by two simple relationships. Acceleration is equal to the change in velocity over the change in time (t here represents time). So a change in velocity in time is an acceleration. Velocity is a change in position over time. These two equations are usually referred to as the average acceleration and the average velocity. That is, they represent a sort of macroscopic measurement of those things. An example of a velocity is you re in your car, you drive for ten 4

5 seconds down the highway; you have obviously changed your position. That is delta x over delta t. Similarly if you accelerate, you have a change in velocity in time. That s delta v over delta t, an acceleration. Or if you put your brakes on, you change your velocity; you slow down. That is also a change in velocity over time, an acceleration also which we often refer to as a deceleration. If an object, for example, a car moves at a constant speed in a constant direction, then it is not undergoing an acceleration. If delta v is zero, no change in velocity in a certain amount of time, then the acceleration is zero. If a car moves ten meters in five seconds, we can then compute its average velocity, v equals delta x over delta t. That is, a change in position in time. In this case, ten meters over five seconds or two meters per second is its velocity. Although to actually be correct, we would have to say a direction associated with that to be a velocity. Two meters per second is actually the speed of the object. Slide #8 Newton s Laws 1. An object at rest remains at rest unless acted on by a force. An object moving at a uniform velocity continues at that velocity unless acted on by a force. 2. F=ma 3. Every action produces an equal and opposite reaction. Now we come to Newton s three Laws. Newton devised these laws about 300 years ago to describe what he observed about motion and they ve come to be accepted as a law because they re so well established. The first of Newton s Laws says an object at rest remains at rest unless acted on by a force. That is, unless a force acts it s going to sit there and not move. An object moving at a uniform velocity continues at that velocity unless acted on by a force. So an object at rest does not move if a force doesn t act; an object already moving continues to move in constant direction at a constant speed unless a force acts on it to change its speed or its direction. That s Newton s first Law. Newton s second Law is an expression of what force means. What he said was F=ma; force equals mass times acceleration. A force, remember, is a push or a pull; if you push or pull on a mass, that mass accelerates and it s simply proportional to the mass and the acceleration. The m that shows up in this equation is what I referred to previously as the inertial mass. That is, it s what appears to give an object inertia; the force is proportional to that mass and that acceleration. Newton s third Law says that for every action there is an equal and opposite reaction. In the case of a rocket shown here, the hot gas being expelled at high velocity downward causes the body of the rocket to go upwards. The action is the gas going down; the reaction is the mass going up. If I pull on a car with a rope, I am the action and the reaction is that the car pulls back on me. Now I may cause that object to accelerate if I exert enough force, but there is a pull back on me equal to my pulling forward. Those are Newton s three Laws. Slide #9 Motion Example [Image] Three plot diagrams showing constant acceleration, velocity, and position. 5

6 Let s consider a motion example where there is a constant acceleration, we will look at a plot of acceleration, a plot of velocity, and a plot of position. Now an example of constant acceleration is you re on a highway and you put your foot on the accelerator so that you evenly accelerate up to some speed. Another example of constant acceleration is acceleration of gravity. Here near the surface of the earth, that acceleration is constant. So in the first curve on the upper left, we plot acceleration versus time and we see that acceleration is a constant therefore it is just a straight line across, horizontal line and the slope of that line is zero. The second plot that we see is a plot of velocity versus time. We start out at some initial velocity, V 0, don t know what that is, but we re starting out at some velocity which could be zero and we see that we accelerate; we increase velocity due to the presence of that acceleration. The slope of that line in the plot is a, the acceleration. That is, the constant acceleration, a, produces a line in the velocity plot with a slope a. Now let s look at the third plot. The third plot is position versus time, x versus time. When we started out at time zero, we had an initial velocity, V 0. The slope of the line there early on there at the beginning of the curve is V 0, whatever that speed is initially and as we continue to accelerate we change our position more and more rapidly so that at some other later time the slope of the line is increased and the slope is whatever the velocity is at that time. In other words, no matter what time it is, the slope of that curve is equal to the velocity that we have at that point in time. So that s the relationship between acceleration, velocity, and position. Slide #10 Equations of Motion a=constant v=v 0 +at v 0 =constant x=x 0+ v 0 t+1/2at 2 x 0 =constant If a=0, then velocity is constant: v=v 0 If a=0, and v 0 =0, then the position is constant: x=x 0 So let s consider the equations of motion that result from constant acceleration. The equations of motion are simply mathematical statements of what the acceleration speed and position are as a function of time. The acceleration, in this case, is constant, some number. The speed then is equal to the initial speed, v sub zero, plus acceleration times time. The position, x, can then be written as the initial position where the object started plus the initial speed times time plus onehalf of the acceleration times the time squared. That equation, you may recognize, is the equation of a parabola and that is the shape of the curve in the x versus time plot. If the acceleration is zero in this example, then we see that the velocity is a constant, v is simply equal to v not and it remains that way. If the acceleration is zero and the initial velocity, v, is zero, then position is constant, x is equal to x not. It stays in that position if acceleration is zero and initial speed is also zero. We can use these equations to determine where an object is at any point in time. Slide #11 Problem Solving 6

7 Problems in physics can be approached systematically o Draw a picture o What is being asked for? o What is given? o What else do you know? o Determine the method o Write out the steps and solve Recall that we talked about the fact that there was a systematic way to approach problem solving. You start off by drawing a picture, you write down what is being asked for, you write down what is given, you write down what else do you know, you determine the method for solving, and then you write out the steps to solve the problem. We are now going to apply this systematic approach to solving a problem. Slide #12 Equations of Motion Example If you fall from an airplane flying at an altitude of 6 miles, how long will it take to reach the ground, if there is no air resistance? What is given? o x 0 =6 miles x 1609 m/mile=9654m o v 0 =0 What else do you know? o v=v 0 +at o x=x 0 +v 0 t+1/2at 2 o a=-9.8 m/s 2 acceleration due to gravity (down) Okay, let s apply this systematic approach to solving an example of equations of motion. Assume you fall from an airplane flying at an altitude of six miles. How long will it take to reach the ground assuming there s no air resistance? Well, what are you given? You re given that you have an initial position x 0 of six miles. We re going to convert that to meters by multiplying by 1609 meters per mile to get 9654 meters. We re going to take our initial speed as zero, our initial speed downward. We are moving forward at a high speed, but we re going to ignore that for the moment; we re only concerned about our speed going downward and that s initially through. So what else do we know? Well, we know those equations of velocity we just saw. The velocity at any point in time is the initial velocity plus the acceleration times time. The position is the initial position plus the initial velocity times time plus one-half the acceleration times time squared. The other piece of information is a new one that I m providing here and that is that the acceleration, in this case, is the acceleration due to gravity and that s 9.8 meters per second squared. Notice that I have a negative sign in front of it. That indicates that the direction of the acceleration is down toward the ground as opposed to up higher in the air. So we re going to use the equations of motion with an acceleration of -9.8 meters per second squared and an initial velocity of zero and an initial position of 9654 meters. Slide #13 7

8 Equations of Motion Example (2) Method: Find time (t) when x=0 Solve o x=x 0 +v 0 t+1/2at 2 o 0=9654m+0+1/2(-9.8m/s 2 )t 2 o t= (2x9654/9.8) o t= (1970) o t=44.4s Less than one minute! How fast will you be moving? o v=v 0 +at o v=0+(-9.8m/s 2 )(44.4s) o v=-435m/s= -973 miles/hour So we want to define how long it would take to hit the ground if we jumped out of this airplane. That is, we want to find the time when x equals zero. We re starting out at some height of 9654 meters; we re going to call the surface of the earth zero so we re going to solve for x equals zero. So we have our position equation: x equals x not plus v not t plus ½ at squared. Let s plug the numbers into it. So we have zero for the final position is equal to 9654 meters (that s x not) plus zero (because v not is zero) plus ½ the acceleration of 9.8 meters per second squared times time squared. So we re going to rearrange that equation to solve for time and time is equal to the square root of the quantity two times 9654 over 9.8 or t equals the square root of Well, that turns out to 44.4 seconds. So you re in an airplane six miles up and in less than a minute you slam into the ground. Now, we ve been ignoring air resistance and we ll come to that in a minute. Let s ask a related question. When you hit the ground, assuming this is true, how fast are you going? Well, we can use the other equation of motion. You re velocity is equal to the initial velocity plus the acceleration times time or the velocity is equal to zero (our initial velocity was zero) plus 9.8 meters per second squared (negative representing the fact that it s downward) times 44.4 seconds or the velocity is equal to 435 meters per second or in terms of miles per hour, that s 973 miles per hour. If there s no air resistance, you d be going really, really fast when you hit the ground. The negative sign here represents the fact that your speed is downward; your direction and velocity ends up being down toward the earth. Slide #14 Equations of Motion Example (3) But, what really happens is that you rapidly reach a terminal velocity of ~100 miles/hour. How long does it take to reach the ground then? What else do you know? o a=0 constant velocity o v miles/h=-45m/s Method: Find time (t) when x=0 o x=x 0 +v 0 t+1/2at 2 o 0=9654m+(-45m/s)t+0 8

9 o t=214s o t=3.6 minutes Let s go back and do this problem taking into account the fact that there is really air present and that it does have a big influence on the answer. So we re going to change this question to you jump out of this airplane six miles up, how long does it take you to reach the ground including air resistance? Well, we re going to solve this problem approximately because what happens when air is present is you start out at zero velocity down and you accelerate until you reach about 100 miles an hour. That s what s called the terminal velocity. That is, you get up to 100 miles an hour and wind resistance is pushing against you preventing you from accelerating further and you then hurtle toward the earth at a speed of about 100 miles an hour at a constant speed. It obviously depends upon the size of the body and whether you put your wings, your arms out or not, and all those sorts of things, but we ll use 100 miles per hour. The approximation you make is that you travel at 100 miles per hour the entire distance. We could solve the problem exactly, but it s a little bit more complicated and is not needed to illustrate the point. So what happens if you move 100 miles an hour downward instead of just keep accelerating? We have in this case a zero acceleration. That is, we are moving at a constant speed and our initial speed, v zero, we re going to take to be minus 100 miles an hour; the minus, again, representing that it s downward toward the earth and that s equal to minus 45 meters per second. If you convert minus 100 miles an hour to meters per second, it s minus 45. The method is the same; we re going to find the time when the position is zero so we start with our equation x equals x not plus v not t plus ½ at squared. Our final position is zero so zero equals 9654 meters (that s our initial position) plus a negative 45 meters per second (which is v not) times the time plus zero because we said there s no acceleration. In that case, you find the time is 214 seconds. That is, 3.6 minutes. So there s a big difference. When there s no air resistance, it took you only 45 seconds. With air resistance, it takes you three and half minutes. You have a long time to think about what s going to happen when you go into the earth, but now instead of travelling 963 miles when you hit you re only going to be going 100 miles an hour when you hit the ground. Now if you re really going to do this problem exactly, including the fact that you re going to accelerate from zero to 100 miles an hour, you d find out it took just about four minutes to fall that six miles. So that s an example of how the equations of motion can be used. Slide #15 Dropping Weights Time for you to perform an experiment: get up and find two objects that have different masses but that you can drop safely wherever you are, e.g., an eraser and golf ball. Do not use something really large and light like a piece of paper you need to keep air resistance low. Stand on a chair with your arms outstretched and drop the objects at the same time from the same height. Which one hits the floor first? Now it s time for you to do an experiment. It s related to the same experiment that Galileo did 9

10 several hundred years ago which was actually done by Guy Steven, but in this experiment I want you to get two objects that are different in mass, maybe an eraser and a golf ball or something like that, two things that are different in mass but about the same size. We re going to do a test where you drop these to the ground. You will not want to use something like a piece of paper because their air resistance is very important. So what I want you to do is take these two objects, stand up on a chair, hold your arms out in front of you with one object in each hand, and drop the two objects from your hand and watch and listen for when they hit the ground. So stand up, stop the lecture, do this experiment now. Okay, which object hit the ground first? Slide #16 Dropping Weights Hopefully you found that the two objects hit the floor at the same time. The reason is in the equation x=x 0 +v 0 t+1/2at 2 There is no dependence in this equation on the mass of the object. There is something fascinating about science; one gets such wholesale conjecture from such a trifling investment of fact. Mark Twain Hopefully in your experiment you found the same answer that Galileo found when he dropped two objects off the top of the leaning tower of Pisa. That is, the two objects should have hit the ground at the same time. The time it takes for the two objects to reach the ground is independent of the mass of the two objects. You could have dropped a bb and a bowling ball and they would have hit the ground at the same time. The basic reason is answered by the equation of motion, x equals x not plus v not t plus ½ at squared. You ll notice that mass does not appear in that equation. The motion depends upon the initial position, the initial velocity, and the acceleration, but there s no dependence on mass. Objects move under that acceleration and those initial position and speed in exactly the same way, independent of their mass. Now this reminds me of a quotation from Mark Twain. He said, There is something fascinating about science; one gets such wholesale conjecture from such a trifling investment of fact. Now I think that science is fascinating because of that. We can start with a few observations and a very small amount of information and imply tremendous things. We can understand the motion of balls falling off the top of the leaning tower of Pisa and use that to describe how planets move around the sun or comets move through cosmos or galaxies rotate and move through the universe. I ll tell you a little story related to this quotation. I used to travel to Russia fairly often to do an experiment in a fairly obscure, southern part of Russia in a little town called Neutrino. The Russians have this habit of sitting around the dinner table and having toasts and everybody has to get up and give a toast so since we were talking about science and stuff I decided, well, I d get up and use this Mark Twain toast, There is something fascinating about science; one gets such wholesale conjecture from such a trifling investment of fact. Well, it wasn t until after I made that statement that I realized that s an exceedingly difficult quotation to translate. Can you imagine a translator trying to get the meaning out of those words across to an audience that doesn t speak English? So I learned something there. A few people did understand the quotation, but it took a very lengthy explanation to get the message across. Slide #17 10

11 Three Dimensional Motion Motion in each of the directions (x, y, z) is totally independent of one another. Instead of releasing two balls and letting them fall to the ground, drop one and throw the other one exactly sideways. When do the two balls hit the ground? Let s turn from one dimensional motion to discussing three dimensional motion. In three dimensions, we typically represent these as x, y, and z coordinates. Z is usually taken as the up and down direction and x and y are the two other orthogonal directions. In this type of motion, typically, we would have gravity acting down along the negative z direction and we d pick one of the others, let s say x, to represent the motion of, let s say, a football that s thrown through the air. So we represent the x motion of the football and the z motion of the football and we picked those coordinates so we don t have to worry about the y direction. We re going to have you do an experiment again with your two balls that you dropped on the ground before. When you dropped them previously, just straight down, the two different masses hit the floor at the same time, at least that s what you d expect to happen. This time instead of dropping both balls, you re going to drop one straight down and at the same time throw the other one to the side. Let s say use your right hand and you re going to throw it to the right. The picture here sort of tries to show that; the yellow ball drops straight down and the blue ball is tossed to the right. If you release them at the same time, the question is when do the two balls hit the ground? Now stop the lecture at this point and go try this experiment before you go on to the next slide. Slide #18 Three Dimensional Motion (2) The two balls hit the ground at the same time! The motion down and the motion sideways are independent of each other. It does not matter how fast you throw the ball sideways, the time to hit the ground is the same as the ball that is dropped. If you conducted the experiment correctly, what you would have noticed is that the two balls hit the ground at the same time. How can that be? The ball thrown to the right obviously travelled a greater distance, but it still hit the ground at the same time as the one just dropped. The reason for this is that the motion in each of the directions x, y, and z are independent of each other. It doesn t matter what the motion in the x direction is, the motion in the y direction is controlled independently. So in this case, gravity acts to pull the balls down and both balls fall toward the ground at the same rate and hit the ground at the same time. This would be true even if you used a rifle. If you had a rifle horizontally facing and you shot the rifle at the same time you dropped a bullet to the ground, the bullet you dropped would hit the ground and at the same time the other bullet that was shot through the air and might have travelled maybe a mile horizontally would also hit the ground at the same time. Those motions are independent of each other. Slide #19 11

12 Motion of Real Objects If you throw a ball through the air, it follows a parabolic curve. Now that we ve briefly looked at three dimensional motion, consider the motion of real objects through the air. If I take a basketball and throw it through the air, it follows the curve of a parabola. The shape of that curve is determined by the fact that there s a constant acceleration downward at the same time that the object is moving in the other dimension. Very simple motion and you ve observed a basketball rotating at the same time as it s flying through the air. That rotation has no effect on the parabola that the ball follows. Slide #20 Motion of Real Objects (2) If you throw a more complex object through the air, its center of mass also follows a parabolic curve. o A thrown shoe will tumble at the same time it follows a parabola. o Try it yourself. Let s now consider what happens with a more complex object. I want you to do an experiment. I want you to take one of your shoes and throw it through the air like you would a basketball. Obviously try not to hit something with your shoe, but watch how that shoe flies through the air. What you should notice is that the center of the shoe follows a parabola through the air, the same shape that a basketball follows while at the same time the shoe tumbles over and over, end to end, and side to side. It rotates and tumbles, but the basic motion is the same. This is true of any complex object; it follows the basic parabolic shape through the air and at the same time it rotates about, what is called, its center of mass. Try some experiments throwing objects through the air and observe what happens. Slide #21 Range Range is the distance that a thrown object reaches. The range of a thrown object depends on the angle at which it was thrown. Objects thrown at 45 degrees have the largest range. One of the characteristics of objects being thrown through the air is the distance that it travels. That is what s called its range. The range of an object is determined, basically, by two parameters. One is the speed with which it is thrown and the second one is the angle with respect to the ground with which it is thrown. The best angle to throw an object, in order to have the greatest range, is at 45 degrees above the ground so if I wanted to throw a football the very farthest I could with my arm, I would throw it at 45 degrees. If I throw the ball at 30 degrees 12

13 above the ground, it will go a shorter distance because, again, it s falling under gravity and it hits the ground sooner than it would at 45 degrees, but the interesting thing is that if I throw the ball at 60 degrees from the ground, that is steeper up in the air, it will always travel the same distance as when I threw it at 30 degrees. The difference between the 30 degree path and the 60 degree path is when I throw it at 60 degrees it will take longer to get to the same place. So if I m a ball player and I m throwing a football, I m going to use the 30 degree angle to get it to somebody who s close because I want to get it to them quickly, but if I want somebody to run a long distance and be there to catch the ball, I might want to throw it at 60 degrees because it will hang in the air longer. On the other hand, I ll throw it at 45 if I want it to go the absolute greatest distance I can throw. So range is a characteristic of motion determined by two parameters, the angle and the speed. Slide #22 Monkey and the Gun Watch the video clip on the monkey and the gun A monkey hanging from a tree sees the flash of a hunter s gun and drops from the tree. The bullet from the hunter s gun aiming at the monkey falls at the same speed as the monkey. There s a demonstration called the monkey and the gun. There s a videotape available for you to watch on the website and in this videotape there s an introduction that I did about motion and range and things like that and then there s a video showing an experiment called the monkey and the gun. The monkey and the gun is a simple demonstration. The story is that there s a monkey hanging from a tree and there s a hunter hunting the monkey. The hunter aims at the monkey and the monkey seeing the flash from the gun drops from the tree because he hasn t taken physics and he thinks he s going to escape the bullet dropping from the tree, but, in fact, what happens is the monkey falling from the tree falls at the same rate as the bullet from the gun falls toward the ground. In fact, the monkey has done the wrong thing by letting go of the tree and bullet hits him. The video will show you this demonstration and you should watch it now, if you haven t already watched it, and then return to this lecture. Slide #23 Complexity So far we have considered simple motion, but lots of details effect actual real life motion. Consider a baseball to determine exactly how far it will go, you need to know: o Initial speed in three dimensions o Air resistance o Gravity and any local variations in it o Spin of the ball o Light pressure on the ball o Temperature of the ball o Roundness of the ball 13

14 So far we have been talking about motion as a simplistic situation. We ve generally ignored air resistance and other effects and just as an example, I wanted to go through and talk to you a little bit more about what happens in real life because if we wanted to predict exactly where an object landed, we have to take into account a lot of other effects. Let s consider, for example, a baseball. The baseball has been hit by a bat and flies through the air. What would we have to know in order to predict exactly where that ball was going to go? Well, obviously, we have to know its initial speed in all three dimensions, that is x, y, and z, and then we d have to take into account air resistance because we know air does effect the motion of that ball. So those two are obvious. Gravity has an impact, but one of our first factors that could be complex is there might be local variations in gravity. This is well known that there are mountains that have high-density rock, iron in them, for example and they locally perturb the gravity so that they cause slight effects on motion. Now we re talking about very, very small effects that make millimeter differences on where the ball ends up, but it is an effect. The spin of the ball effects where it goes because air is dragged around the ball and causes the ball to curve as it s flying through the air so we have to know about its spin and, again, that s related to the presence of air. If we get down to the really, really small effects, light reflecting off a ball actually has an impact and moves the object or exerts a force on the object and changes very, very slightly where it would go. The temperature of the ball could have an effect because if the ball is hot from being hit, it warms the air around it as it travels and that could affect the air resistance also. And other factors like how round the ball is, if it s a little bit out of round that could affect, again, how the air might flow over it and where it goes. So the point of this is to figure exactly where something goes, we have to know a lot of factors. On the other hand, physics does allow us to compute all of these factors and to determine exactly where an object goes to within a range of the atomic level. When we get to the atomic level, we will see that there are other factors that influence how things move. Slide #24 Friction Friction is caused by roughness of surfaces at the microscopic level rubbing against each other. Friction at rest (static friction) is greater than friction in motion (kinetic friction). The frictional force always acts in the direction to oppose an external force. Friction depends on the mass and shape of an object. [Image] Diagram of friction and external force on mass We are all familiar with the effect of friction. That s the resistance of one object moving over another one such as something being dragged across the floor; there s a resistance to that motion. Resistance is caused by microscopic roughness of one object against another and that tends to cause resistance to that motion. An object being moved over a very smooth surface, like ice, moves easily. If you try to drag that same object over sandpaper, it doesn t move very easily. There are two types of friction. One is called static friction and the other kinetic friction. Static friction is the force of when an object is standing still and you re trying to make it start to move. 14

15 Kinetic friction is the friction that s observed once the object has started to move. In general, static friction is larger than kinetic friction. That is, it s harder to get an object to start moving than it is to keep it moving. Again, this has to do with the way in which things interact at the microscopic level. Friction is a force that always acts in the opposite direction to an external force. If you connect rope to a block and try and drag it across the ground, you re pulling in one direction, friction of the force that s in the other direction and friction will grow and grow and increase in size equal to and opposite to the external force being applied until you reach the point that you break that static friction and the object then starts to move. In general, once you started it moving, the kinetic friction is slightly less and you can keep it moving more easily than you could get it started. Friction does depend on the mass of an object; the heavier an object in a given space, the harder it is to move. That is, the greater the friction and the shape affect it. The smaller the footprint that mass is exerted over, the more the frictional force. Slide #25 Topics for this Lecture Topics for This Lecture Mass Force Motion: acceleration, speed, and displacement Newton s three Laws Equations of motion One-, two-, and three-dimensional motion The monkey and the gun Complexity versus approximation Friction In this lecture, we ve looked at a number of topics dealing largely with mechanics. We ve looked at what is mass, what is force, we ve looked at motion, acceleration, speed, and displacement, we ve talked about Newton s three Laws, the equations of motion, one and twodimensional motion, we ve talked and observed the monkey and the gun experiment, we talked a little bit about complexity, and finally we introduced friction. This is a quick overview of mechanics of motion which is a whole subject area in physics and will form the basis of some of our future discussions. 15