fzx physics lecture notes reserved fzx: personal...

FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved

FZX: Personal Lecture Notes from Daniel W. Koon

St. Lawrence University Physics Department

CHAPTER 4

Please report any glitches, bugs or errors to the author: dkoon at stlawu.edu.

4. Force Newton’s First Law Newton’s Second Newton’s Third Law Newton’s Second Law and Circular Motion Applications of Newton’s Laws Drawing your Free-body diagram Causes (Weight, Normal, Tension, Friction) More specific examples (Inclined Planes, Pulleys) Digging Deeper: static and kinetic friction

page 1 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


FZX, Chapter 4: FORCE We have spent the last few chapters doing ‘kinematics’, the study of HOW things move. Now it is time to do some ‘dynamics’, the study of WHY things move. Dynamics is the whole point of mechanics: if you can’t explain why things move or predict how things will move, your knowledge of motion is not very useful. We do know that in order to get something to move that isn’t moving, we do have to give it a shove or something, so let’s start there. The first definition we must make is that of ‘force’. A force is something that can change the motion of an object. A force can make a stationary object move, or bring a moving object to rest, or change the direction or magnitude of the velocity. But it needn’t change the motion of the object. What do I mean by that? Well, if I am involved in a very closely matched tug-of-war, I may be exerting a force, but, since it is countered by my opponent’s force, there is no change in motion. This is why I say that.... ....a force is something which CAN change the motion of an object. [Definition of force] If I pull on a rope with some force, and someone else pulls in an opposite direction with the same magnitude of force, there is no net effect, It is as if there was no force acting at all. This tells us two things: first, we had better pay attention to the vector nature of forces (What if we had been pulling in the SAME direction?) and second, that it is the ‘net’ or ‘total’ force that counts. [Definition of net force] ...21 ++=∑ FFF

rrr

The definition of a force as something that MAY change the motion of an object might not seem a very useful definition to you. It is, after all, an operational definition. If it doesn’t fit your needs, how about this operational definition: A force is something which can be measured on a bathroom scale. Equivalently, we can say that.... ...a force is something that can be measured in ‘pounds’. [Alternate definition of force] (Don’t dismiss this one yet: we will use this definition in the near future!) The MKS units of force are called ‘Newtons’, which are defined as N = 1 kg.m/s2 = 1lb/(4.45) [Units of force] NEWTON’S FIRST LAW: Let’s turn our first definition of force around and say that... ...if there is no force acting on an object, its motion will not change. [Newton’s First law] Uh-oh, you’re thinking: we all know that things slow down if left unattended, and that to keep something moving, you need to supply something. That’s where you’re wrong, but you’re in good company. The reason things slow down when left unattended is precisely because the net force on them is not zero. Friction is the culprit. The First Law is often split into two parts. An object at rest remains at rest unless acted on by a nonzero external force [Newton “1A”] An object in motion remains in motion with a uniform speed and direction unless acted on by a nonzero external force. [Newton “1B”] The first of these is not very controversial. It’s the second one that we are accustomed to not believing. To verify that nature does in fact work according to Newton “1B”, let’s try to get rid of friction. If you are driving a car on a rocky road, the vehicle slows down quickly once you take your foot off the pedal. If you move the car onto a paved road, it slows down less rapidly. If you put it on an icy road, it may hardly slow down at all, even if you apply the brakes. If you could eliminate friction entirely, the car would travel at a constant speed unless you applied some other force to it. Since it is impossible to completely remove friction, you must resort to this ‘thought experiment’ in which you consider the limiting case of no friction. If this did not make sense, keep thinking about it. This is one of those instances where your common sense may refuse to budge. Try to come up with counterarguments to that bit I said about objects not changing their motion unless an outside force acts. Then see if you can shoot down your counterargument. If not, discuss it with classmates or the instructor. If you do not struggle with this material, it will probably not sink in.



NEWTON’S SECOND: Okay, so what if there IS a net force acting on an object? We would expect the object’s motion to change, but how? The velocity will change, so there will be an acceleration, and the acceleration will be proportional to the net force. The acceleration will be inversely proportional to the mass (also known as ‘inertia’) of the object. We write this law, Newton’s Second Law, as [ Newton’s Second Law ] amF rr

=∑ Look at this equation this way: the left hand side tells you what causes the change in motion, and the right hand side tells you what the effect of the force was. Repeat the following mantra: "Left side: Cause; Right side: Effect". The left hand side is NOT a new force. It is merely the result of all the forces that are acting on an object. We need to find all of these forces before we can say anything about what effect they will have on the object of interest. To analyze a problem that requires Newton’s Second Law, you will want to draw a picture which shows all the forces acting on an object. Such a diagram is called a ‘free-body diagram’ or an ‘isolation diagram’. There are probably other names as well. ‘ ‘ or ‘m ‘ will not be one of the forces you draw on your diagram, because it is just the sum of the forces. To add it to the diagram would like adding your checking account balance to your deposits while trying to balance your checkbook: you would be counting the same money twice. Make sure you don’t “cook the books.”

∑Fr ra

We will discuss specific forces and talk about how to set up an FBD in the last half of this chapter.

NEWTON’S THIRD LAW: There are some laws of physics which have become such an integral part of our cultural heritage, so that, even if we only vaguely understand what they mean, we can still recite them in our sleep. Newton’s Third is one of these. It is most commonly stated, ‘For every action there is an equal and opposite reaction.’ [ Newton’s Third Law ]



But what does it mean, particularly in terms of physical quantities, and in terms of force, which is what we are knee-deep in in this chapter? ‘Action’ is an older word that has been used to mean ‘force’. We still refer to ‘action at a distance’ to refer to a force exerted between two objects that are not touching. So we could rephrase Newton’s Third to say, ‘For every force there is an equal and opposite reaction force.’ [Newton’s Third Law] If this makes more sense to you, use it instead of the formulation we gave above. It is vital in identifying forces to associate every force with the particular object which feels that force. In analyzing a force problem, it is essential to identify one object for which you wish to catalogue all the forces acting on it. That’s what ∑F

r

means: the sum of all forces acting on ONE particular object. If I consider the forces acting on me as I stand on the floor, there will be a weight, pulling down on me. There will also be a contact force exerted on me by the floor. How do I know this? Consider what my motion is if the floor is not there. Now, if I consider the floor, it also experiences a contact force, one due to my presence. Newton’s Third Law simply says that this force -- the force on the floor due to me -- is equal and opposite to the contact force on me due to the floor. Newton’s Third is put most eloquently, I feel, by Paul Hewitt, who writes ‘You cannot touch without being touched.’ [ Hewitt’s version of Newton’s Third ] The biggest problem people have understanding this law is that they lose track of which force is acting on which object. People tend to think ‘equal action and reaction: don’t they cancel?’ These two forces, which are equal and opposite, DON’T cancel because they act on different objects. If you and a classmate find yourselves stranded in the middle of an ice hockey rink, and you want to leave the ice, you can try pushing on each other. Draw a sketch of the two of you, and for each of you, label the forces acting. Neither of you will have a nonzero net y-component of force, because each weight will be balanced by the ‘normal’ supporting force from the ice. All that is left are the ‘action’ and ‘reaction’ forces each of you exerts on the other. If these cancelled, the two of you would stay stranded at center ice, but instead each travels off in an opposite direction across the ice. If you alone were stranded on the ice, with nothing to push off of, you would have to count on the measly amount of friction you can get from the ice. (Maybe not so measly, considering that hockey players seem to be able to push off the ice itself quite easily)



NEWTON’S SECOND LAW AND CIRCULAR MOTION: We have said that Newton’s Second Law -- amF rr

=∑ -- is about cause and effect: the left side of the equation shows the cause of the observed motion, and the right side shows what effect the forces will have on the motion. Which side do we know about when we are dealing with centripetal (i.e. circular) motion? Let me leave that question in the air a few seconds while I remind you that we said that centripetal acceleration is . This means that the centripetal FORCE is

rvac /2=

r

mvmaF cc

2

== . [Centripetal force]



The question I posed in the last paragraph becomes: Do I put Fc on the lefthand side of my amF rr=∑ equation, or do I

put ac on the right hand side? Obviously I can’t do both, or I get an equation that isn’t the least bit useful. The answer is, I put ac on the righthand side, because ‘centripetal’ means that I observe a particular type of motion, not that some new force called the ‘centripetal force’ is at work in my problem. It is the effect I observe, not the cause of said effect. This is an important point that is often missed by students: there is no such independent force called the centripetal force that is similar to frictional, tension, or normal forces. In fact, any one of these might contribute to the centripetal motion. ‘Centripetal’ only DESCRIBES the EFFECT of whatever force or forces cause the object to travel around in a circle. One important cause of a centripetal effect is gravity. The Earth travels around the Sun, and the Moon travels around the Earth, because of gravity. In these cases, we cannot use ‘W=mg’ to quantify the force. Our expression for the weight is a specific instance of a far more general expression, which is

2rGMmF = , [Gravitational force]

where ‘G’ is a constant, 1.67 x 10-11N.m2/kg2, ‘m’ is the mass of the object feeling this force, ‘M’ is the mass of the object pulling on the object of interest, and ‘r’ is the distance between the CENTERS of the two masses. Near the Earth’s surface, . 22 /8.9/ smrGMg == By the way, although we are pulled by gravity from the Earth, the ground also exerts a normal force upward on us, so that the ‘centripetal force’ that turns us around the Earth’s axis is not just the gravitational force, but the vector sum of it and the normal force.



APPLICATIONS of NEWTON’S LAWS Newton’s Laws, particularly the Second Law, provide a very practical way of analyzing the motion of a system consisting of more than one object. The power of the laws rests in the fact that we can consider each object separately, just as we can analyze two-dimensional motion by considering x-motion and y-motion separately. The way we separate the objects from each other is through the free-body diagram [FBD]. DRAWING YOUR FREE-BODY DIAGRAM: There are a few steps to constructing an accurate FBD: First, isolate the object of interest. If there is more than one object of interest, draw a separate FBD for each. If the two objects are in contact, it is HIGHLY recommended that you redraw at least one of the objects elsewhere to avoid a confusing, overcrowded diagram. If at any point in the procedure your diagram gets overcrowded, redraw it, perhaps using one diagram to label forces, a second diagram to label dimensions. Need I remind you that it’s vital that your diagram not be confusing, especially to you? Second, draw a dotted line around the object of interest. This is so that later you can be dead certain that you know all of the objects in contact with it, so that you can be confident that you have considered all of the forces acting on the object. Third, draw the forces. This should go in two stages: First, consider all non-contact forces, or ‘actions at a distance’. So far in this course, gravity is the only one we’ve considered, so that simplifies this step. With noncontact forces out of the way, all that’s left are contact forces. Use your diagram to locate all the objects touching your FBD object. Next, draw the corresponding contact forces. It is a good idea for now to draw a small dot in the center of your object, and to draw all of the vectors as originating from there. Fourth, write down the ∑F

r equations. Remember, since Newton’s Second Law is a vector equation, it is a shorthand

for TWO scalar equations. Write



∑ = AND xx maF ∑ = yy maF across from each other on your paper. On the next line, write down all the appropriate force components, as well as anything you might know about the acceleration, especially if it is zero. All that’s left now is the algebra. Good luck! To implement the program I’ve outlined above, you need to know how to identify forces. Remember that I’ve referred to Newton’s Second as a Cause--Effect equation: the lefthand side lists the causes of the observed acceleration, and the righthand side lists the effects of the observed forces. There are four types of forces we shall consider in this section in constructing a free-body diagram or ‘FBD’. We will also list some of the types of effects you are likely to encounter. CAUSES: WEIGHT: , is an ‘action at a distance’. This is the only force we will deal with this semester that acts between two objects that do not touch (the Earth and the object being considered). The weight is proportional to how much ‘stuff’ there is in the object that experiences weight. We call this quantity the ‘mass’, which has MKS units of kilograms, or ‘kg’. Note that even when you feel weightless, like when you are falling toward the Earth in free-fall, there is still a weight acting on you, and it has the same magnitude -- W=mg -- and the same direction -- straight down -- as it ever did.

rW

NORMAL: , is a ‘contact force’, as are all the other forces we shall consider for now. Its being a contact force is convenient, because when we are making an inventory of all the forces acting on an object, we first consider Weight, the only action-at-a-distance that we know, and then we consider all the other forces, which must therefore be contact forces. So all we have to do is look to see what things are touching the object of interest. The normal force acts perpendicularly (‘normally’) to the surface between two objects in contact. It is the force that the ground exerts as it keeps you from falling through the floor. In THIS case, the Normal force exactly equals the Weight, but this is not something you can always rely on. There is no fixed formula for the magnitude of the normal force: you are either given it, or you will need to calculate it. By the way, there IS a way to measure the normal force: put a bathroom scale under the object you are looking at. The scale will read the normal force.

rN

There is another way of interpreting the normal force: it is sometimes called the ‘apparent weight’. Consider riding an elevator. As the elevator starts moving upwards, you feel heavier. A bathroom scale underneath you would register a larger than normal reading for your weight. The scale is NOT measuring your weight, which, as we said above, is just equal to ‘mg’ and does not vary during your trip. The scale is measuring the normal force between you and the floor. Since this force does not equal your weight, that means that the net force, ∑F

r, is not equal to zero, so you must be

accelerating. What does a bathroom scale underneath you read if you are falling out of the second-floor window? What normal force is acting on you?



rTENSION: T is a contact force exerted by a string -- or rope, chain, or similar item -- which is pulling (or pushing) on the object. Again, there is no formula for the magnitude of this force, but -- if the string is massless -- it will always point in the direction of the string. Sometimes you will be given this quantity, as in a problem where you are given the maximum tension the string will allow before breaking, but otherwise, this will be another unknown in your algebra. THERE IS NO UNIVERSAL FORMULA for tension. If there is more than one piece of string in a problem, you will have more than one tension. Use subscripts to distinguish between them. On the other hand, if a single piece of string is hung around a pulley, then -- if it is a massless string, and physicists usually assume this to be the case -- the tension will be uniform throughout.



FRICTION:

rf is a contact force that can exist at the surface between two objects. It will be exerted parallel to the

surface, and will point in the direction in which it resists other forces’ attempt to move the surfaces relative to each other. If the surfaces are actually moving, it will point opposite the direction of motion. If the surfaces are not moving, then imagine which way they would move if there were no friction. Imagine spreading grease or oil between the two surfaces: this will tell you which way the objects would move if there were no friction. Friction will point opposite this direction. For example, if you are standing in the aisle of a bus as it starts to speed up from rest, which way does the frictional force point? It will point in the opposite direction from the direction you will move -- relative to the bus -- if the floor is so slippery that the bus can’t tug you along in its direction. In this example, the friction actually causes your motion, by pointing in the direction which would oppose your moving RELATIVE to the surface you’re in contact with. Quantifying friction is tricky. Friction is like the normal force, it is an intelligent force. The surface between two objects will exert just enough friction to keep them from slipping. How do these objects know how much friction to exert without exerting too much? Ya got me. In both cases, however, there are limits. If you jump on a table, it calculates -- in no time at all -- just how much of a normal force to exert to keep you from accelerating through to the floor, without exerting so much that it tosses you up into the air. But, if you try to drop an elephant on the same table, chances are that the table can’t exert that large a force, and so it exerts what it can as it collapses. Same with friction. If you push on this table from the side to get it to budge, friction knows enough to push back sufficiently strongly to counter your push, but not so strongly as to get the table to move opposite to the direction you’re pushing in. Once you have pushed the table beyond the limit that friction is willing to exert, then a frictional force is still exerted, but one that is equal to the maximum that the table will exert. This value is fmax = μN [ Maximum friction exerted ] where ‘N’ is the normal force, and ‘μ’ -- lower case Greek character mu -- is called the coefficient of friction, and is USUALLY between 0 and 1.



To summarize: CAUSES: FORCE MAGNITUDE DIRECTION r Weight, W mg straight down

Normal, No formula perpendicular to the surface of contact rN

Tension, rT No formula along the rope, string, chain, etc.

Friction, rf ≤μN parallel to the surface of contact

EFFECTS: IF AN OBJECT STARTS AT REST.... If ∑F

r=0, then =0, and it stays at rest.

ra

If ∑Fr

is not zero, but =0, then the object will start to move in the direction of rv ∑F

r.

IF AN OBJECT IS IN MOTION... If ∑F

r=0, then =0, and the velocity will be constant.

ra If

ra is parallel to , then the object will speed up. rv

If r ra is opposite v , then the object will slow down.

If ra is always perpendicular to

rv , then the object will travel in a circle. MORE SPECIFIC EXAMPLES: There are a number of applications of these forces that occur a lot in physics, and that are also useful for checking whether you know how to apply Newton’s Second Law. We will discuss a couple of them. INCLINED PLANES Consider a car parked on the street that is not flat. What are the forces acting on it? Draw a sideview of the car on a street at about 22o (Draw a 45o angle and halve it.) tilt, compared to the horizontal. Mark the center of the car with a dot: we’ll draw all the forces as originating at this dot.



Actions-at-a-distance? There’s the weight acting on the car, pointing straight down. Draw it. Contact forces? There’s a normal force, because if the street weren’t there, the car would fall. The normal force points perpendicularly to the surface between road and car, so it points at 22o from the vertical. Draw it. Now these two forces can’t cancel out, since they don’t point in exactly opposite directions. If the car is parked well, so that it doesn’t budge, then there must be some other force at work. Since the only things touching the car are the road and the air, and since the air is not keeping the car from sliding, there must be a frictional force at work. If there were no such force, which way would the car accelerate? Down the hill. This means that the frictional force is up the hill. Draw it. Okay, let’s quantify. We want the three forces to add up to zero: r

+r

+r

=0. This means that the x- and y-components must both add up to zero. There are two questions here: what ARE my x- and y-axes, and what are the components of each of the forces? What follows is important for doing any problems on inclines, so don’t just read it: work through it, making sure to draw it!

W N F

Since the motion of the car would be either up or down the slope, we choose the x-axis to be along the surface of the street. This means that, even if the car would start to move, there is no acceleration along the y-axis. We DO NOT use the traditional horizontal and vertical directions for our x- and y-axes. Sketch the x-axis along the direction of the street and the y-axis pointing in the direction of the normal force, perpendicular to the road’s surface. Label both. I gave away what the components of the normal are. Since it is in the y-direction, we have Nx = 0 Ny = N. As for the friction, since it points up the hill, fx = f fy = 0. It is the weight that poses the biggest problem. Usually (when we use the vertical direction as the y-axis) the weight points in the negative-y direction. But now it has both an x- and a y-component. Sketch the weight pointing down. Now break it down into its x- and y-components. This means drawing a right triangle for which the vector

r is the hypotenuse.

Inspecting this triangle, we see that r

W is at an angle θ to the y-axis, which is the same angle by which the street rises

above the horizontal. Since the direction of r

is measured against the vertical, rather than the horizontal, as is usually

done, the components for r

are

W

WW

Wx = -Wsinθ Wy = -Wcosθ. This is the one major exception to the rule that associates x-components with cosines and y-components with sines, and it only works because the angle θ is measured relative to the vertical. Make a note of it. For any object that sits on an inclined plane, the normal and weight will need to be considered. There may or may not be friction: you will probably be given that information. There may be other forces as well. The point I want to make is to show you how to set up the axes and what the components of

rN and

rW are.



PULLEYS Throw a rope around a pulley, put a mass on one end and pull on the other end. What’s going on in terms of forces? We

can think of the rope as an ingenious invention that allows us to transfer a force from one end to the other, sometimes even changing the direction in which the force is applied. Draw a rope around a pulley with a mass hanging down from one end and someone pulling on its other end. Now draw a free-body diagram of the mass. The force exerted on it by the string will equal the force with which you pull the other end, but it will point up, rather than down, which is the direction that the rope pulls. The mass will, of course, have a weight pulling down on it. The motion of the mass will depend on how these two forces acting on it compare. If you pull with a force greater than the weight of the mass, the mass will accelerate upwards. If you force is less than the weight, it will accelerate downward. If the two forces match, it will not accelerate: if it is at rest, it will stay at rest, and if it is moving, it will move with a constant velocity. Let’s consider the forces acting on the rest of the system. Consider first the rope. We usually employ massless rope in physics problems. It’s a lot easier to deal with conceptually than real rope is. For anything that is massless,

mra =0, whether it is accelerating or not. This means that the total force on the rope must be zero.

Draw another sketch of this problem, but sketch an FBD of the rope. It is being pulled downward on both ends by the mass and by you. These two forces are equal, and are equal to the tension, rT , in the rope. This means that in order for the net force on the rope to be zero, there must be an upward force of magnitude 2T somewhere. The only thing that can provide this is the pulley. Now let’s consider the pulley. It’s supporting the string, which is pulling down on it with a total magnitude of 2T. It must be supported by something else which has a magnitude of 2T in order for it not to accelerate. So there must be some supporting structure holding the pulley up. (A pulley in mid-air, which is not supported, is not much use.) Draw an FBD for the pulley itself. The pulley problem can be complicated if the rope on either side is at different angles. Just remember the following... A single rope exerts equal forces of magnitude T on objects connected to either end. The rope exerts a force of magnitude T on either side of the pulley. The net force on the rope must be zero.



DIGGING DEEPER (Static and Kinetic Friction): If you’ve ever noticed that an object seems to become easier to push once it starts moving, you might want to distinguish between ‘static’ and ‘kinetic’ friction. The observation I just mentioned would suggest that the difference between these two types of frictions is that the static coefficient of friction is greater than the kinetic coefficient -- ‘μ’ is smaller once you start moving. We can use subscripts to distinguish between them: ks μμ > . Since static friction may be less than the maximum, but kinetic friction is always the max, we can write




NfNf

kk

ss

μμ

=≤

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