Fizyx for Felines: A Physics Textbook for the Curious Cat
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Skona BrittainSun, 30 May 2010 07:14:00 +0000enhourly1http://wordpress.com/https://s0.wp.com/i/buttonw-com.pngFizyx for Felines: A Physics Textbook for the Curious Cat
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Inter-Chapter Interlude
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https://fizyxforfelines.wordpress.com/2010/05/28/inter-chapter-interlude/#respondSat, 29 May 2010 06:46:38 +0000http://fizyxforfelines.wordpress.com/?p=407Last week there was no post because my computer was still feebly recovering from a power outage. Next week there will be no post because I’ll be taking a group of high school mathletes to a competition in Las Vegas. And this week there is a special post that is not a section of a chapter. So the question of whether Chapter 3 will ever get written is still unresolved. This excerpt is a historical tale about a magnificent cat named Magnus who had a historically significant tail. Not only is there no math in it, but there’s very little physics in it.
Fizyx For Felines
Inter-Chapter Interlude – Magnus
…behind every great human there is an even greater cat.
Magnus (1596-1965) was a magnificent Scottish cat with two uncommon characteristics. He had an almost-canine enjoyment of accompanying his people on long walks, not from behind in a backpack like Norton, but along the ground and in all directions. His other unusual trait was a flair for physics. Fortunately, he had the opportunity to spend each of his lives with a physicist with whom he could contribute to the advancement of the field.
Magnus was very proud of his fine Scottish name and indeed spent almost all of his lives with Scottish physicists, half of whom bore the common Scottish name John. However, many historians of science feel that he did his most important work during the only life he spent outside of Scotland, his middle life. Since that work is the subject of the next chapter, this note will focus on his other lives.
His first life was spent with John Napier (1550-1617), a physicist, mathematician, astronomer and astrologer, who actually did more lasting work in mathematics than physics. He was the inventor of logarithms, which allowed the complicated operation of multiplication to be replaced by the simpler one of addition, and also of a tool called Napier’s Bones, an alternative to the classic abacus. This latter invention was instigated by Magnus claiming the household abacus as a cat toy. He would delight in rolling the round beads along the wires with his paws whenever John was trying to do a calculation. Hence, Napier’s Bones used fixed straight rods instead of movable round pieces. See Figure i-1.
Figure i-1. Napier’s Bones.
After those accomplishments, along with raising a dozen kids, John began indulging his non-scientific side and acting increasingly weirdly. He was suspected of dabbling in black arts, and of using a black rooster as a “familiar spirit”, which is a demonic role more typically played by black cats. Magnus, who was almost completely black himself, became spooked by this behavior, and vowed to stay away from mathematicians in the future. Historians have wondered why Magnus attributed Napier’s questionable sanity to his pursuit of mathematics, but the correctness of his intuition was affirmed by a close friend during his eighth life, Georg Cantor’s cat.
The final act between Magnus and his first person was the pigeon incident. A neighbor’s pigeons wandering onto the Napier grain field so annoyed John that one day he got rid of them by poisoning them. This indirectly poisoned his beloved pet Magnus, who, ironically, would have been happy to eliminate the pigeons for John himself.
Thus ended Magnus’ first life. All the rest of his lives were spent with physicists rather than mathematicians, and it was several lifetimes before he deigned to associate with even a mathematical physicist.
His second life was spent with Joseph Black (1728-1799), a Scottish physicist and chemist, whom he found to be an exceptionally affectionate man, possibly because Joseph never had a wife or children to siphon his affection away from Magnus. Joseph founded the field of thermochemistry, a precursor to thermodynamics, with his 1761 discovery of latent heat. Henceforth, thermodynamics was Magnus’s favorite area of physics. He especially appreciated participating in experiments involving warmth and soft pressure.
In his next life, Magnus lived with John H.D. Anderson (1725-1796), who didn’t become a physicist until rather late in his life, in the 1760s. This John always had a strong pedagogical bent, and devoted himself to bringing popular science applications to underrepresented populations, such as the working class. To this end, he freely gave non-academic evening lectures, at which, in a radical break from the conventions of 18th century Scotland, he welcomed women, and even felines. After wandering into one of these events as an inquisitive young kitten, Magnus followed John home to further pursue the ideas presented, and John graciously obliged him. During the last decade of his life, John produced five editions of a physics textbook, Institutes of Physics, which Magnus scrupulously edited.
Magnus’s person in his next life, Sir John Leslie (1766-1832), also modified his career in mid-life, but in this case it was away from Magnus favorite area of concentration, the study of heat, to a more mathematical endeavor, at which point Magnus left him.
Even within the subject of thermodynamics, this John had a tendency to go in the opposite direction from Magnus’s inclinations. His interest in exploring the absence of heat led him to be the first person to artificially produce ice, which he did with an air pump in 1810. Magnus more fondly recalled John’s 1804 experiments with radiant heat. The legacy of their time together includes the creation of an object that John dubbed Magnus’s Cube, but that the rest of the world knows as Leslie’s Cube. It’s a cubic container with surfaces of different colors and texture (see Figure i-2). It can be filled with either cold water to study absorption or warm water to study emission. Magnus liked to directly engage himself in emission studies by curling up on the dark side.
Figure i-2. Leslie’s Cube.
Magnus’s happiest life was spent with Michael Faraday (1791-1867), an English experimentalist, who happened to be the least mathematical of all of Magnus’s physicist people. However, it was not the man himself but rather his other cat, Electra, who contributed the most to Magnus’s bliss. As mentioned above, you’ll hear more about them and their dynamic relationship in the following chapter.
After Electra’s death in 1849, in a halfhearted attempt to return to Scotland by suicide, Magnus next aligned himself with the Scottish physicist James Clerk Maxwell (1831-1879), who was living in Cambridge at the time and extending Magnus’s previous person’s work. Although James is best known for the equations that bear his name, which are central to Chapter 3, he also made extremely significant contributions in both optics and kinetic theory, the latter of which were inspired by Magnus and will be featured in Chapter 4.
His seventh life was spent with John James Waterston (1811-1883?), a Scottish physicist specializing in Magnus’s favorite field of thermodynamics. Although this John did good work on the kinetic theory of gases, it was not appreciated during his lifetime. One day, in mid-June of 1883, the two of them went for one of their customary walks, from which John never returned and was presumed dead.
On the other hand, although he was also presumed dead then, Magnus did return, in his next lifetime, when he became the companion of James Dewar (1842-1923), a Scottish chemist and physicist. In 1892, he inspired James to invent the insulated bottle still known as the Dewar flask, the precursor to the modern thermos (see Figure i-3). James’s original motivation was the desire to keep Magnus’s cream insulated during their long walks. But he also found the flask to be very useful for transporting low-temperature gases for experiments in atomic physics.
Figure i-3. Dewar’s Flask.
Peter Higgs (b.1929), currently an emeritus professor of theoretical physicist at the University of Edinburgh, was homeschooled as a child due to his asthma, which unfortunately also precluded ever having a pet in the house. So Magnus had to confine his relationship with Peter, his last person, to accompanying him on outdoor walks. Fortunately, Peter loved to walk, and it was on one of their walks together, in 1964 in the Cairngorms, a mountain range in the Scottish Highlands, that he claimed he had his “one big idea”, now known as the Higgs mechanism. The Higgs mechanism is a broken symmetry theory in particle physics, which is beyond the scope of even Chapter 9 of this book. However, the main idea of the broken symmetry was undoubtedly, albeit perhaps subconsciously, inspired by Magnus’s anti-symmetric antics on the trail.
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https://fizyxforfelines.wordpress.com/2010/05/14/chapter-2-excerpt-8/#respondFri, 14 May 2010 20:26:27 +0000http://fizyxforfelines.wordpress.com/?p=398This is the final excerpt of the chapter, which finally explains the chapter’s title (Newton’s Cat’s Kittens).
Summary and Application
We summarize the rotational ideas discussed in this chapter alongside their translational analogs of the previous chapter in Figure 2-27, an expansion of the table in Figure 2-1.
Figure 2-27.
[Linear] Displacement (s)
Angular Displacement (θ)
[Linear] Velocity (v)
Angular Velocity (ω)
[Linear] Acceleration (a)
Angular Acceleration (α)
Force (F)
Torque (τ )
Mass (m)
Rotational Inertia (I)
[Linear] Momentum (p = mv)
Angular Momentum (L = Iω)
Kinetic Energy: ½mv^{2}
Rotational Kinetic Energy: ½Iω^{2}
Newton’s 2nd Law: F = ma
τ = Iα
Reformulation of 2nd law: F = dp/dt
τ = dL/dt
Conservation of [Linear] Momentum
Conservation of Angular Momentum
Finally, we will bring all these concepts together by applying them to a situation that you probably encounter daily – the cat flap door. (We realize we are assuming that the reader is neither homeless nor captive. The first assumption is justified by the price of this book; but for those of you whose loss of freedom is captured by the euphemistic label “indoor”, we suggest that you request your person buy you a cat flap door for the noble purpose of conducting physics experiments. )
~ Historical Note ~
Not only did Isaac Newton’s cat inspire his musings about angular momentum conservation, but she played a pivotal role (pun intended) in one of his most enduring inventions – the cat flap door.
After completing his development of classical mechanics, Newton began focusing (pun not intended) on optics. His resultant theories of the properties of light will be discussed in Chapter 4. Here we are more interested in how he managed to conduct his explorations of light in a dark room. To his dismay, his cat kept pushing the door open in the middle of an experiment, letting in light that interfered with the results. A lesser person might have firmly shut the door to prevent such occurrences, but Isaac had more respect for his cat. He understood that the mere presence of a closed door is a source of frustration. Thus, necessity being the mother of invention, he created the world’s first cat flap door, by cutting a hole in the door and hanging a piece of dark felt over it.
Speaking of motherhood, when his cat had kittens during this time period, she naturally brought them into the optics room, too. A lesser scientist might have allowed the kittens to go through their mom’s large door, but Isaac dotingly cut out several smaller cat flap doors next to the original one. Although British professional animal writer Pauline Dewberry has claimed this was indicative of the typical genius’s blind spot – that “it didn’t occur to him that they could use the existing one”^{7}, there is no evidence of this lack of thought. If it hadn’t occurred to him, one small door would have sufficed; the fact that he built several indicates that he was just trying to make their lives easier.
Rather, we suspect that it didn’t occur to Pauline Dewberry that rotational physics shows it would actually be much easier for the kittens to use the smaller ones! In fact, as we will derive in the text, they would have had to work four times as hard if Isaac hadn’t made them their own small flaps. Furthermore, his work probably made them feel special!
To this day, cat flaps are more popular in Great Britain, the homeland of their inventor, than in the United States. Between 88% and 92% of British cats have access to the outdoors.^{8} Interestingly, this corresponds quite precisely to the approximately 90% percent of pet cats that are non-pedigreed, although they are not necessarily the same set. Whereas in the U.S., many more pet cats are forced to be indoor cats, and many more are pedigreed. Apparently the pilgrims’ quest for increased liberty did not apply to their pets.
Your cat flap door may be a lot fancier than Newton’s cats’ doors, but the basic principle is the same. Whether the hinges are metal or the hinging effect is achieved by an adhesive, the axis of rotation is the one through the top of the flap. You probably know from experience that the further away from the axis you push, the easier it is to rotate the door. So your door should be mounted high enough that the bottom is at nose-height, not paw-height.
Figure 2-28.
It is most effective to apply the force perpendicular to the door. In this case, the moment arm is the distance from the axis of rotation to the point where the force is applied. However, if the force is being applied tangentially, only the component perpendicular to the door will cause any rotation. Then the moment arm is the shorter distance from the axis of rotation to the line along the direction of the force.
Even when you start pushing the door with your nose at the efficient right angle to the door, if you continue to push straight ahead while you walk through, that angle changes, because the door rotates to different positions. See Figure 2-28. Thus the torque decreases as you proceed with constant force.
Note that after you have passed through the door, the flap continues to rotate back and forth, in accordance with Newton’s Third Law, like a pendulum. Friction dampens the motion, causing it to eventually halt, which is good because that keeps the raccoons out.
Now let’s calculate the effect of flap size, as alluded to in the historical note above. Figure 2-29 illustrates a large flap door for a mother cat and a small cat door for her kitten, similar to the situation in Newton’s household.
Figure 2-29.
If the original felt piece was rectangular with height H and mass M, the moment of inertia of the mother cat door would be (1/3)MH^{2}, as was derived earlier in this chapter. Assuming that the kittens’ flaps were half the height and half the width of their mom’s flap, their mass would be a quarter of its mass, and hence their rotational inertia would be only one-sixteenth that of their mom’s flap: I_{kitten}= (1/3)(M_{kitten})(H_{kitten})^{2} =(1/3)(¼M)(½H)^{2} = (1/16) (1/3)MH^{2} = (1/16) I_{cat}
Although twice the angular displacement would be required for the same linear displacement at the bottom, and twice the force is required to achieve the same torque, there is still a factor of four remaining. This shows why the kittens would have had to work four times as hard if they’d had to use their mom’s door.
In later chapters we will see how these simple flap doors have been greatly enhanced by applications from other areas of physics, such as magnetism and optics.
]]>https://fizyxforfelines.wordpress.com/2010/05/14/chapter-2-excerpt-8/feed/0skonabrittainChapter 2 – Excerpt 7
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https://fizyxforfelines.wordpress.com/2010/05/07/chapter-2-excerpt-7/#respondFri, 07 May 2010 15:43:02 +0000http://fizyxforfelines.wordpress.com/?p=390Yet another short excerpt, including a physics approach to analyzing catfood.
A Highly Recommended Lab Exercise
We rarely even suggest very involved laboratory experiments, as we generally prefer the thought experiments of theoreticians to the actual ones of experimentalists, but the following has significant redeeming value. We think that by performing it, you have a lot to gain, dietetically as well as pedagogically.
It is a well-known fact that the more expensive the canned catfood, the moister the contents. In your previous lives, it was easy to evaluate quality by price. But now that price tags have been replaced by bar codes, how can you determine the moisture content without opening a can? This experiment will enable you to rank cans of catfood by their levels of moisture.
Lab Exercise – Predicting Catfood Liquidity
Gather several different kinds of canned catfood and set up an inclined plane. For the plane, we particularly recommend a rectangular cat scratching board. If you don’t already own one, tell your person you need one for a scientific experiment. The kind with embedded catnip is particularly well-suited to our purpose, as will be explained below. (And afterwards, you can scratch it and enjoy the embedded catnip.)
Conduct individual trial runs as follows. Place a can at the top of the inclined plane on its side and let go. Be careful to not add any external force by pushing it, so that its initial speed is zero. Using a stopwatch, time how long it takes for it to reach the bottom of the plane.
Note that the longer the runs, the easier they are to time and obtain statistically significant timing differences. The length of time can be increased two ways: increasing the distance, i.e. the length of the board; or decreasing the speed of the rolls. In turn, the speed can be decreased two ways: decreasing the angle of elevation; or increasing the coefficient of friction. The best way to achieve a high coefficient of friction is to use a board with a rough texture. That’s why we recommended a cat scratching board, and the embedded catnip may add even more texture. If necessary to slow the motion down further, just adjust the angle by lowering the top of the board.
Now that you know which can rolls down the fastest and which the slowest, you need to interpret your results to determine which is best. Note that the more liquid the contents are, the more they will slosh around within the can, rather than just rolling along with it. Thus you can determine, from their relative speeds, which can contains the moistest, hence most delectable, treats.
Hint: If you are having trouble analyzing the data theoretically, add a can of evaporated milk to the experiment. This is an example of a general technique in both mathematics and theoretical physics, as well as in experimental physics: It is often useful to consider extreme values, or boundary cases, whenever deriving principles.
After all that work, you deserve to eat at least one can of catfood. Select the one that you have experimentally determined to be the best-tasting one and ask your person to open it for you. For extra credit, convince your person to open all of the cans so that you can verify all of your results.
If your person is hesitant to open many cans, here’s some extra ammunition. Say that you are also studying the can opener and wish to observe how the torque applied to the rotating handle leads to the rotation of the can in a different plane. Many instances of observation will be required to fully comprehend this phenomenon, even for your person, who will, we hope, become intrigued enough by the can opener’s operation to repeatedly engage it. If so, be sure to give credit to its inventor, William W. Lyman of Meriden, CT, who patented it in 1870.
The kinetic energy of rotation for an extended object is found by adding up the kinetic energy amounts of the individual pieces of rotating mass Δm. From Chapter 1, we know that they each contribute an amount of ½(Δm)v^{2} to the sum. Applying equations (2) and (5) yields the rotational analog of this expression:
KE = Σ½(Δm)v^{2} = Σ½(Δm)(rω)^{2} = ½[Σ(Δm)r^{2}]ω^{2} = ½Iω^{2}
Motion can be separated into its translational and rotational components, with each part analyzed separately, and then the results combined to produce the actual motion. For example, consider the motion of a spinning ping pong ball: the center of mass moves in a parabola while the body rotates around the center of mass. Thus the total kinetic energy is the translational kinetic energy of the center of mass plus the rotational kinetic energy around the center of mass, i.e. KE_{total} = KE_{trans} + KE_{rot} = ½mv^{2} + ½Iω^{2}
Energy considerations provide the simplest way to calculate the speeds obtained by rolling objects, just as they did for purely translational motion. At the end of chapter 1, we used this approach to calculate the speed of a freely falling object when it hit the ground, which was, of course, independent of its mass: v = (2gh)^{½}, where h was the height where it started at rest. Now consider a rounded solid object, such as a sphere or a cylinder, rolling down a ramp of height h. Assume its initial speed at the top, is zero and that it rolls without slipping down the ramp.
Since only shape determines the rotational inertia per mass, all solids of a particular shape, regardless of their size or mass (as long as their density is uniform) will roll down with the same acceleration due to gravity, and hence the same speeds. This is analogous to the fact that all objects undergo free fall with the same acceleration due to gravity. However, it may be surprising because the rotational inertia does depend on size, being proportional to the square of the radius, and hence is greater for larger objects. It turns out that this leads to differences in angular speed, but not linear speed, as the following energy calculation shows.
For uniform solid cylinders, the rotational inertia is I = ½MR^{2},
so the rotational kinetic energy is KE_{rot} = ½Iω^{2} = ½(½MR^{2})ω^{2} = ¼M(RW)^{2} = ¼mv^{2}
and hence the total kinetic energy is KE_{total} = KE_{trans} + KE_{rot} = ½mv^{2} + ¼mv^{2} = ¾mv^{2}.
At the top of the ramp, all the energy is potential energy and when the bottom is reached it’s all kinetic. So, relying on conservation of energy, we set our expression for the final kinetic energy equal to the initial potential energy and then solve for v: mgh = ¾mv^{2} → v = ((4/3)gh)^{½}.
Suppose instead that the cylinder were hollow and very thin. To make the calculation much simpler, let’s assume it has neither top nor bottom and hence more symmetry, such as a catfood can with both the top and bottom cut off. Then it’s basically a thin hoop, with I = MR^{2} and we have
KE_{rot} = ½Iω^{2} = ½(MR^{2})ω^{2} = ½M(Rω)^{2} = ½ mv^{2} KE_{total} = KE_{trans} + KE_{rot} =½mv^{2} + ½mv^{2} = mv^{2} and mgh = mv^{2} → v = (gh)^{½}.
All the cases in the lab experiment suggested above are between these two extreme cases of a solid cylinder and an empty cylinder, with the cheaper brands of catfood being closer to the former case than the moister brands.
Q. A uniform solid ball rolls down an inclined plane of height h, starting from rest, without slipping. What is its linear speed the moment it touches the bottom? A. KE = KE_{trans} + KE_{rot} = ½mv^{2} + ½Iω^{2} = ½mv^{2} +½((2/5)MR^{2})ω^{2} = ½mv^{2} + 1/5 M(Rω)^{2} = ½mv^{2} + 1/5 mv^{2} = 7/10 mv^{2}. So PE_{initial} = KE_{final} → mgh = (7/10)mv_{final}^{2 }→ v_{final}^{} = ((10/7)gh)^{½}.
Note that all the speeds being obtained are a fraction of the final speed for free falls from the same height. They’re slower because some of the energy is being taken up by the spinning. And although the linear speeds are independent of size, the smaller objects are spinning faster when they reach the bottom, since v = rω.
Furthermore, these results are also independent of the angle of inclination of the ramp. For a shallower inclined plane, the trip will just take longer. The same final linear speed is gained more slowly, due to the smaller component of gravitational force directed down the ramp.
]]>https://fizyxforfelines.wordpress.com/2010/05/07/chapter-2-excerpt-7/feed/0skonabrittainChapter 2 – Excerpt 6
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https://fizyxforfelines.wordpress.com/2010/04/30/chapter-2-excerpt-6/#respondSat, 01 May 2010 06:06:52 +0000http://fizyxforfelines.wordpress.com/?p=368Last Friday was the first one on which I didn’t post an excerpt since this project began, 14 weeks ago. The reason why is that I was unusually busy, because a major project (an opportunity to sell my math clocks at a festival) was suddenly superimposed on an already very full weekend. However, no matter how busy I am, I am sufficiently obsessive-compulsive that I would have squeezed it in, except that I completely forgot about it. Ironically, this lapse of memory was probably due to lack of sleep, which in turn was due to my staying up late to watch a movie (because it had to be returned that morning) that – here’s where the irony comes in – was about a literately successful blogger, namely Julie and Julia. Perhaps hearing about her attentions from vast quantities of blog readers caused me to repress thinking about the contrast with my small set of readers and even smaller set of commenters. And even when I did belatedly remember this project, I was not inspired to get back on track.
So I really want to encourage comments on the blog. For those of you who know me, who keep giving me comments in person or by email, please put them on the blog instead. And for those felines among you, I realize it’s hard to type when your paws are wider than the keys, but know that I really appreciate your efforts.
Meanwhile, I am making up for this lapse it by including a double excerpt today, with lots of pictures, including a couple of cartoons. Although today is National Hairball Awareness Day, I cannot think of any way to relate that to the topics in today’s excerpt. Maybe that’s a good thing – there are no gross hairballs forthcoming. Rather, you will see graceful ice skaters and ballet dancers, both human and feline, and learn some of the secrets behind their motions. Also revealed are the related but more complex directions for the even more graceful maneuvering required for you to always land on your feet.
Moments of Inertia of Extended Irregular Objects
Your body has three principal axes of rotation. They are the three most symmetric mutually perpendicular axes through your center of mass (see Figure 2-18(a)). Human bodies also have three such axes (see Figure 2-18(b)). In both cases, it is easiest to rotate about the longitudinal axis, because the mass is concentrated closer to it. Human acrobats do spins, flips, and cartwheels around their longitudinal, transverse and median axes respectively, in order of increasing difficulty due to increasing moments of inertia.
Figure 2-20. The moment of inertia is smallest around the longitudinal axes. For felines this axis is horizontal, while for humans, it’s vertical.
The longer your legs, the greater your rotational inertia, the harder it is to run quickly. This you find it easier to run fast than a horse or giraffe does but more difficult than a dachshund or a mouse. Regardless of their leg length, anyone can always reduce their rotational inertia by bending their legs. Thus, as we run, we naturally tend to bend our legs, since we can tell that it’s a lot harder to swing them when they’re straight.
Sometimes you actually want to increase your rotational inertia, such as when you are walking along the top of a thin fence. To maintain your balance, you need to keep adjusting your center of gravity. Your tail comes in very useful for this, giving you extra resistance to rotational motion and hence allowing you more time to adjust. Since human tightrope walkers don’t have tails, they hold long poles instead to obtain this increased stability.
Consider a spinning ice skater or ballerina. (See Figure 2-21.) By extending her arms, she can triple her rotational inertia, and by extending one of her legs as well as both arms, she can increase it six-fold.
Figure 2-21. The skater on the right has even more than six times as much rotational inertia around her longitudinal axis, because she also is extending her torso to the side.
Angular Momentum
Angular momentum, symbolized by the non-Greek letter L, is the rotational analog of linear momentum, which was defined as mass times velocity; hence it is defined analogously as the rotational inertia times the angular velocity:
(6)
L = Iω
Thus the angular momentum of an object depends on how the mass of the object is distributed around the axis of rotation, as well as the rate at which the object is rotating. Q. What is the angular momentum of the earth about its axis of rotation? A. Combining the results of the previous exercises in this chapter, we have L = Iω ≈ (9.7 ×10^{37} kg m^{2})x(7.3×10^{-5} rad/s) ≈ 7.1 kg m^{2}/s.
Recall the principle of the Conservation of [Linear] Momentum: if there is no net external force on a system, then its total amount of momentum is a constant. Analogously, there is an equally momentous principle of the Conservation of Angular Momentum, which was also developed by Isaac Newton. It states that if there is no net external torque on a [rotating] system, then there will be no change to its angular momentum.
Mathematical Digression
Mathematically, this principle is just a reformulation of the rotational analog of Newton’s Second Law. Note that just as force is the time derivative of linear momentum, torque is the time derivative of angular momentum. And, to be ultra-precise, angular momentum is actually the moment of momentum, and hence should be defined as L = r x p. It follows thatL = Iω holds in scalar form but the vector equation (6) holds only for symmetrically rotating objects, which are the only kind ever studied in physics classes. In those cases, L and ω point in the same direction. Since L must be constant, it obviously cannot be in the constantly changing direction of motion, but must, in fact, be perpendicular to it, as given by the Right Paw rule.
People often take advantage of the principle of Conservation of Angular Momentum for recreational purposes, such as ball games. See Figure 2-22. It explains why spinning footballs are more stable than non-spinning ones.
Figure 2-22.
Above we mentioned that a dancer could increase her moment of inertia six-fold by extending three of her limbs. Alternatively, if she starts a spin with the three limbs extended, and then pulls them all in, this move decreases her moment of inertia six-fold. By the Conservation of Angular Momentum principle, her rotational speed, or spin rate, must simultaneously increase by the same factor of six, which looks very impressive (see the schematic figures in Figure 2-23(a) and the authentic one in Figure 2-23(b)).
Figure 2-23. (a) The skater on the right is spinning 3 times as fast as the one on the left. (b) The English choreographer Christopher Wheeldon demonstrates a low moment of inertia pose. (c) Loki, a “male ballerina”, lives in New York City with his photographer Odd Todd.
Although cats engage in similar recreational activities (see Figure 2-21(c)), the primary feline application of the principle of Conservation of Angular Momentum is far more serious, sometimes even life-saving.
Practical Application
How to Always Land on Your Feet
At first thought, it may appear that, if a nasty human drops you head-first without any spin, you would land head-first on the ground, since, by the principle of conservation of angular momentum, you cannot create spin where there is none. Once let go, you are effectively an isolated system, so your angular momentum is conserved. However, there are ways of getting around this predicament, so to speak. Since there are no external forces on you other than gravity, you must exert internal forces to accomplish your task. You have probably already learned how to do this in a previous course, if not from your mom in kittenhood, but, just in case, the technique will be described here.
You need to execute a sequence of twists and turns without ever changing your angular momentum. Thus, if you turn one part of your body one direction, you need to simultaneously turn another part in the other direction. Such a turn is called a zero-angular-momentum rotation. However, if that was all you did, you wouldn’t make any net progress. In order to keep from making a series of contortions that just undo each other, you need to repeatedly modify your moment of inertia.
To be explicit, one method is to first pull in your legs and tail while arching your back, then turn your head and legs through a large angle toward the ground while rotating your torso slightly in the other direction. Then extend your legs and tail and straighten your spine, so that your torso can catch up to your legs while they backtrack only a small angular distance. Repeat if necessary. For additional assistance, use your stiffened tail as a propeller, and cushion your landing with all four paws. See Figure 2-24.^{5}
Figure 2-24. This is a time-lapse photo sequence of one falling cat during a fraction of a second, not seven cats.
While you presumably find these instructions to be quite clear, non-feline animals are not as adept at following them. See Figure 2-25.
Figure 2-25. (a) Far Side cartoon by Gary Larson, February 23, 1988 (b) Garfield cartoon by Jim Davis, September 21, 1989.
Humans also don’t have this skill, but in their case it’s more understandable. Being normally upright creatures, their principal axis of rotation about which it is easiest to twist is one that has no effect on changing their orientation with respect to the ground. And they are significantly less agile, partly because they have only twenty-five vertebrae, five fewer than felines. Furthermore, they don’t have the advantage of tails. Thus, although humans can perform similar maneuvers, they cannot do so fast enough to avoid landing head-first if they are dropped head-first. However, professional astronauts, after much practice, have managed to attain the skill of zero-angular-momentum twists about each of their principal axes in zero-gravity environments.
On the other hand, some inanimate objects manage to achieve what canines, rhinos and humans cannot. For example, according to Murphy’s Law, buttered toast always lands buttered side down. This lead us to suggest, but not recommend, the following experiment.
Lab Exercise – Testing Murphy’s Conjecture
To test whether your survival instinct is powerful enough to supersede Murphy’s Law, strap a piece of buttered toast to your back, then try falling from various heights. Note that it should be buttered side up, so that you and it have the same preferred orientation. Hence, you won’t get your fur oily from doing this laboratory work. However, if Murphy’s Conjecture turns out to be correct, not only will the floor get oily, but you may sustain bodily injury. Thus, we suggest performing it on very plush carpet and, in the event of such unfortunate results, hiding well when your person discovers the mess.
While many people know Newton discovered gravity from watching an apple fall, far fewer realize that he discovered the conservation of angular momentum by watching his cat fall. He noticed that Spitface always landed on her feet and by repeated observation he deduced how she conserved angular momentum while doing so.
As recent studies have shown, survival from a fall depends on height in a surprising way. Cats falling from the seventh floor have a survival rate approximately 30% lower than that of cats falling from the 20th floor. Apparently this is because it “takes about eight floors for the cat to realize what is occurring, relax and correct itself.” ^{6} Thus, had he lived in the modern age of skyscrapers, Descartes may have concluded that cats do have souls!
]]>https://fizyxforfelines.wordpress.com/2010/04/30/chapter-2-excerpt-6/feed/0skonabrittainChan on Ping Pong tableChapter 2 – Excerpt 5
https://fizyxforfelines.wordpress.com/2010/04/16/chapter-2-excerpt-5/
https://fizyxforfelines.wordpress.com/2010/04/16/chapter-2-excerpt-5/#respondSat, 17 Apr 2010 02:40:06 +0000http://fizyxforfelines.wordpress.com/?p=339A mathematical excerpt, using integration to calculate moments of inertia of inanimate shapes, such as a cat flap door.
Moments of Inertia of Extended Regular Objects
Extended objects can be considered to be composed of infinitely many point masses. Perhaps more easily we can imagine a very large number of very small masses. Let’s call each tiny chunk of mass Δm. To calculate the total rotational inertia of an object, we need to add up the contributions from all the pieces of mass:
(5)
I = Σ Δmr^{2}
To be accurate, we actually need to take limits, letting the tiny pieces of mass become infinitesimal pieces approaching size zero; this requires the calculus technique of integration – I = ∫r^{2}dm – because the distance r may not be constant for a whole piece Δm, no matter how small it is. However, for simple objects with simplifying approximations, we can perform the addition without using calculus.
For example, for a hoop of mass M with a thickness that is very small compared to its radius R (see Figure 2-14), we can approximate the distance of each piece of mass from a perpendicular axis through its center by R. Hence, each piece of mass Δm contributes an amount equal to ΔmR^{2} to the hoop’s rotational inertia. Adding up all the pieces of rotational inertia just involves adding up all the Δm’s, which have a sum equal to the total mass of the hoop, M (i.e. ΣΔm = M). This gives a total moment of inertia of MR^{2} for the hoop.
Figure 2-14. A thin hoop of radius R and mass M has a moment of inertia about its axis of symmetry equal to MR^{2}.
For an object of uniform density ρ (the Greek letter rho is typically used for density), we have ρ = M/V and Δm = ρ ΔV for each piece of volume ΔV. In two-dimensional cases, these would be ρ = M/A and Δm = ρ ΔA, and in 1-dimension they’re ρ = M/L and Δm = ρ ΔL. Using infinitesimals, these become dm = ρ dV, ρ dA or ρ dL, and we have I = ∫∫∫r^{2}ρdV or ∫∫r^{2}ρdA or ∫r^{2}ρdL.
As a more interesting example, which does require calculus, let’s calculate the moment of inertia of a cat flap door. Say that the flap is constructed of a rectangular piece of material, such as rubber, of uniform density ρ, and has a total mass of M, a height of H, and a width of W. Then ρ = M/A = M/(WH). Putting the origin at the upper left corner of the flap (see Figure 2-15), the axis of rotation is the positive x-axis, so the distance of a piece of mass dm from the axis is (-y). So we have dm = ρ dA = (M/WH) dA = (M/WH) dxdy and hence
Figure 2-15. For a uniform rectangular sheet about its top, I = 1/3 MH^{2}.
Note that rotational inertia is not an intrinsic property of an object. It is only defined with respect to an axis of rotation. For example, let’s compare the moment of inertia of a stiff rectangular flag around its flagpole (see Figure 2-16(a)) to that of one around an axis through its center in its own plane (see Fig 2-16(b)). To illustrate these different axes, we have arbitrarily chosen a team flag and a holiday flag, respectively.
Figure 2-16. (a) The flag of the Northwestern University athletic teams (all known as the Wildcats) (b) A version of this Halloween flag with a more aesthetic aspect ratio is available from flags.com.
In both cases we will put the origin where the axis of rotation meets the flag’s lower edge, with the x-axis along the lower edge of the flag, and the y-axis through the axis of rotation. So r = |x|. Then we have
vs.
This result makes intuitive sense. We would expect the rotational inertia for the former to be four times as large as the latter because, on the average, the mass of the Wildcats flag is twice as far away from its axis of rotation as the mass of the Halloween flag is from its axis.
For round objects, these calculations are easier to do with polar coordinates. As an example, let’s calculate the moment of inertia of a cylindrical can of catfood about an axis through the center of its circular top and bottom. See Figure 2-17.
Figure 2-17.
We’ll call the height of the can H, its radius R, and its mass M. Then with the origin at the center of the bottom of the can, we have Note that this result is independent of the height of the can.
For a sphere, around an axis through its center, it turns out that the rotational inertia formula is I = 2/5 MR^{2}. Q. What is the moment of inertia of the earth about its axis of rotation? A. Since the quantity is being squared, we need to use a more precise value for the radius than we did in the previous chapter. By the above formula, we have I = 2/5 MR^{2}≈ (2/5)(6.0 × 10^{24}kg)(6.37×10^{6}m)^{2}≈ 9.7 ×10^{37} kg m^{2}.
Imagine you are unwinding a spherical ball of yarn of radius R by pulling on it with a constant force F along the floor, as shown in Figure 2-18:. Its rotational speed will increase with a rotational acceleration of . As it gets unwound, R decreases, of course. But the torque τ decreases linearly with R, whereas the moment of inertia I decreases as the square of R. So the rotational acceleration increases inversely proportional to R. (We must admit that this example is oversimplified: as the yarn unwinds, the point of application of your pulling force would vary wildly rather than remain at the bottom of the ball.)
All of the above moment of inertia formulas, along with a few other common ones, are tabulated in Figure 2-19 for reference. Note that the more symmetric cases tend to have a lower rotational inertia.
Figure 2-19. Some Common Moment of Inertia Formulas
]]>https://fizyxforfelines.wordpress.com/2010/04/16/chapter-2-excerpt-5/feed/0skonabrittainChapter 2 – Excerpt 4
https://fizyxforfelines.wordpress.com/2010/04/09/chapter-2-excerpt-4/
https://fizyxforfelines.wordpress.com/2010/04/09/chapter-2-excerpt-4/#respondSat, 10 Apr 2010 00:16:39 +0000http://fizyxforfelines.wordpress.com/?p=332Another short excerpt, with a high concentration of feline toys and habits, including a cartoon
Applications of Torque
For example, if you are sitting on a seesaw that is in an exactly horizontal position, then the gravitational force on you is perpendicular to the radial vector, which is along the board, so the magnitude of the torque is just Fd. Suppose you weigh 10 pounds and your person’s daughter weighs 30 pounds and the two of you are perfectly balanced on a seesaw. To keep it from rotating, you must be sitting exactly 3 times as far away from the fulcrum at the center as she is sitting. If you are sitting any closer than that, the seesaw will tilt so that you are higher up. Such a moment has been captured in the sculpture shown in Figure 2-10.
Figure 2-10. This handmade Russian sculpture is from Ronley’s Jewelers in San Francisco.
A given force can be made more effective at producing torque by increasing either the length of the lever arm or the sine of the angle. The former is the theory behind long-handled tools, such as wrenches, hammers and axes. Holding them at the end of the handle while twisting or swinging them is most effective. And of course it is also important to direct the tool’s motion at a right angle to the handle, maximizing the value of sinθ. A contrasting situation is illustrated in Figure 2-11, where the longer the fork the greater the effort required to hold the piece of meat up, to balance the torque exerted by gravity.
Figure 2-11.Pork^{4} Fork Torque: The bigger & wilder the cat, the longer the fork, the greater the torque, the more the effort.
Q. Two cats are playing with a tennis ball of radius 8 cm. Simultaneously, the smaller cat applies a force of 1 newton at the top edge of the ball while the larger one pushes it in the opposite direction three times as hard at a point 3 cm. above the floor. See Figure 2-12. Describe the motion of the ball around an axis through the center of the ball parallel to the floor and perpendicular to the page. Which cat had the greater effect on the ball’s spin?
Figure 2-12. A force diagram for analyzing torque.
A. The torque due to the smaller cat is rF = (8cm.)(2 N) = .16 Nm., since the force is perpendicular to the radial vector. For the larger cat’s force, the angle from the radial vector to the force vector is the supplement of θ, which has the same sine of (r-3)/r = 5/8. So that torque is (8 cm.)(3 N)(5/8)= .15 Nm. Thus the smaller cat has a slightly larger effect on the ball’s spin and the ball will rotate clockwise around the given axis.
Moment of Inertia
An object rotating or revolving about an axis tends to keep rotating or revolving about that axis, unless there is a change to the force. Note that unlike Newton’s First law, this is not a statement about what happens in the absence of a force. It’s about what continues to happen in the presence of the same continual force.
We saw in Chapter 1 that mass, or inertia, is a measure of an object’s tendency to resist changes to its motion. The resistance to changes in rotational motion possessed by an object is called its rotational inertia, or moment of inertia. It is symbolized by I. (Note that the uppercase ninth letter of the Greek alphabet, iota, appears identical to the uppercase ninth letter of our alphabet.)
Linguistic Note
As an aside, just to break up the monotony of this discussion, consider the misnomer “cat nap”: Although humans know that cats nap extensively, they use the phrase “cat nap” to mean a brief period of rest. Perhaps “moment of inertia” would be a better term for this phenomenon. See Figure 2-13.
Figure 2-13. Catnapping is not a momentary phenomenon.
The rotational inertia of an object depends not only on its mass but also on the distribution of that mass. The further a piece of mass is from the axis of rotation, the more it contributes to the rotational inertia. This makes intuitive sense because it takes more torque to get it to revolve through the same angular displacement.
We would like the rotational analog of Newton’s Second Law to hold. Since the analogs of F,m and a are, respectively, τ, I, and α, the analog of F = ma would be
τ = I α.
Let’s determine what the rotational inertia I would have to be in order to make this true. Using a = rα and τ = rF yields I = τ/α = rF/(a/r) = (F/a)r^{2} = mr^{2}. Thus a point mass m at a distance r has a rotational inertia of mr^{2}.
The moment of inertia is proportional to mass, and proportional to the square of distance. Therefore, if the mass of, say, a thin hoop is doubled, so is its moment of inertia; but if its radius is doubled, then its moment of inertia is quadrupled.
4 – The item pictured is actually a piece of tofu., not only to keep this book kosher but also to keep the handler safe from an overeager attack. <return-to-text>
]]>https://fizyxforfelines.wordpress.com/2010/04/09/chapter-2-excerpt-4/feed/0skonabrittainChapter 2 – Excerpt 3
https://fizyxforfelines.wordpress.com/2010/04/02/chapter-2-excerpt-3/
https://fizyxforfelines.wordpress.com/2010/04/02/chapter-2-excerpt-3/#commentsFri, 02 Apr 2010 22:17:03 +0000http://fizyxforfelines.wordpress.com/?p=323This short excerpt is just about cross products, including the well-known Right Paw Rule, which humans call the Right Hand Rule.
Cross products
Actually, the torque vector is thecross product of the radial vector and the force vector. Since cross products are typically studied in 2^{nd} year calculus, you probably are not familiar with them yet. If tensors made you feel tense, cross products may make you feel cross, but don’t worry – they’re much simpler.
There are two ways to multiply vectors – the dot product and the cross product. The operator symbols • and x are written in bold to indicate that they are operations on vectors. The dot product yields a scalar, i.e. a number, whereas the cross product yields another vector.
The magnitude of the cross product is the product of the magnitudes of the two vectors and the sine of the angle between them. This English mouthful may be easier to take in as an equation: | V_{1} x V_{2} | = | V_{1} | x | V_{2} | x sin θ (see Figure 2-8).
Figure 2-8.The vector cross product.
The direction of the resultant vector is perpendicular to the plane of the two vectors being crossed. Of course, there are always two perpendicular directions, for example up and down. Which of the two possible directions is given by the canonical Right Paw Rule, as follows:
Mathematical Definition The Right Paw Rule for Vector Cross Products
Clench your right front paw as if to knead something. If you curl your toes in the direction from V_{1} to V_{2} , then the direction of V_{1} x V_{2} will be away from the carpal pad or towards the dewclaw. See Figure 2-9.
For those of you who are polydactyl^{3}, there is an easier version of the right paw rule: As you move from V_{1} to V_{2}, V_{1} x V_{2} will be pointing in the direction of your extra claw.
Figure 2-9. A right forepaw, bottom view.
Note that this rule means that the direction depends on the order of the operands. The cross product is not a commutative operation. In fact, V_{2} x V_{1} = –V_{1} x V_{2}.
Q. What are î x j, j x k and î x k ? A. Since î, j and k point in the positive x, y and z directions, respectively, of a rectangular coordinate system (see Figure 1-? or 2-9½), they’re all perpendicular to each other, so sinθ is always 1. And being unit vectors, they all have magnitude 1. So we just apply the right paw rule to get î x j = k, j x k = î, and î x k = – j. (Actually, the way a three-dimensional Cartesian coordinate systems is set up is that the direction of the z-axis is always taken to be the one that makes îx j = k hold.)
Figure 2-9½.The unit vectors in a 3-dimensional rectangular coordinate system.
Thus, recalling expression (3) above, we have
τ = r x F
Note that the torque τ is perpendicular to both r and F, and hence it is perpendicular to the entire plane of motion. So the direction of the torque is always along the axis of rotation. Assuming that that axis is vertical, the torque points upward when the induced rotation or revolution is counterclockwise, and downward when it’s clockwise.
Although these quantities are vectors, we can often just use scalars instead, because the rotation can be in one of only two directions – clockwise or counterclockwise – with respect to the axis of rotation. By convention, the counterclockwise direction is considered positive and the clockwise direction negative, as mentioned above.
3 – Polydactyl is the technical term for the colloquial “double-pawed”, which is obviously a misnomer, since you are all most likely quadruple-pawed. Polydactyl cats are also known as “six-finger cats”, “thumb cats”, or “mitten kittens”. <return-to-text>
]]>https://fizyxforfelines.wordpress.com/2010/04/02/chapter-2-excerpt-3/feed/2skonabrittainChapter 2 – Excerpt 2
https://fizyxforfelines.wordpress.com/2010/03/26/chapter-2-excerpt-2/
https://fizyxforfelines.wordpress.com/2010/03/26/chapter-2-excerpt-2/#respondSat, 27 Mar 2010 04:00:20 +0000http://fizyxforfelines.wordpress.com/?p=311The next excerpt from Chapter 2 is fairly mathematical. It introduces the concepts of centripetal force and torque.
Whenever an object is revolving at constant linear speed v in a circle of radius r, its linear acceleration a has magnitude
(3)
a = v^{2}/r
and direction toward the center of the circle. If you’d prefer to placidly accept this expression rather than see how it’s derived, just skip the following box.
Mathematical Derivation
Figure 2-5. The displacement and velocity vectors are shown at two different times:
displacement at first time
velocity at first time
displacement at second time
velocity at second time
With the center of the circle as the origin, the displacement vectors are radial, and the tangential velocity vector is always perpendicular to the displacement vector. Δs is the change in the displacement vector (colored yellow) and Δv is the corresponding change in the velocity vector (colored purple). Both triangles shown are isosceles triangles with the same angle θ between their equal sides, so they are similar triangles. Hence we have Δv/v = Δs/r or Δv = v/r Δs. Since this is true for all Δs and Δv, we have a = dv/dt = lim Δv/Δt = lim (v/r Δs) /Δt = v/r lim Δs/Δt = v/r v = v^{2}/r. It should be clear that in the limit as Δv goes to 0, the (aqua and magenta) velocity vectors approach each other and their difference, Δv, becomes perpendicular to them and thus points toward the center of the circle.
Note that although angular velocity is actually a vector, we have so far only been considering its signed magnitude, the angular speed. For example, equation (2) does not say that v = rω but rather that |v| = r|ω|. This is because the direction of the angular velocity, as well as the direction of the angular displacement, is very non-intuitively taken, by convention, to be perpendicular to the plane of the motion, along the axis of rotation. When we discuss forces, we will examine this issue more.
Centripetal Force
Since rotational motion is not uniform in direction, we know, by Newton’s First Law, that a force is needed to maintain it. Such a force is called a centripetal force. “Centripetal” means “center-seeking” in Latin. A force in the opposite direction is called “centrifugal”, which means center-fleeing.
These adjectives describe the direction of a force, not its source or nature. Any force that is continually perpendicular to the instantaneous direction of motion is a centripetal force. If it’s the only force present, it causes circular motion of its object, with the center of the circle being the fixed point toward which the force is directed. Thus the gravitational pull of the earth on the moon, which causes the moon to revolve around the earth, is a centripetal force, as is the force exerted by the rope on a tether ball in play, or by a whirling elastic string on an attached toy mouse.
Imagine that the toy mouse’s string were to suddenly break (since it is a more likely occurrence than the earth’s gravitational field suddenly disappearing, and can be made even more likely if you claw at it). Then the mouse would fly off on a tangent. This motion would not be due to a centrifugal force but rather to the absence of the centripetal force. Without the string’s pull on the mouse, the toy would continue to move at a uniform speed in a straight line, in the direction it was instantaneously going at the moment of breakage.
If you have ever watched your person make pizza, perhaps you have seen him repeatedly toss the dough into the air, spinning it. He was not doing this just to tantalize you or to show off, nor was it due to the same instinct that inspires your pre-prandial spinning tosses of a dying mouse. The purpose of the spinning tosses of pizza dough is to evenly spread the dough out into a thinner crust. This spreading out appears to be due to a centrifugal force, but again, the outward motion is an attempt by each piece of dough to continue on a tangential path, which is being combated by the pull of the more inner pieces of dough on it.
As in the above examples, it is generally the case that what appears to be a centrifugal force is actually the unexpected absence of a centripetal force, so centrifugal forces are sometimes called fictitious forces. However, in the frame of reference of someone in the revolving object, such as a bug trapped inside a tether ball, the centrifugal force appears real.
You may have experienced this trapped phenomenon while being driven to the vet. If the car taking you to the vet is rounding a curve toward the left, you will be flung toward the right side of the car. Within the frame of reference of the car, you would tend to feel as if a centrifugal force were acting on you, but from out on the road it’s clear that the road’s centripetal force acting on the car is merely not acting on your body.
Similarly, in Zepto Chan’s favorite toy (see Figure 2-6), the outer wall exerts a centripetal force on the revolving ball, forcing it to stay on its circular path.
Figure 2-6.The Crazy Circle ® cat toy holds a ball in a plastic tubular track with paw-sized openings. We think it’s fun to both use and watch.
Whenever a point mass m is revolving at speed v in a circle or radius r, the centripetal force on it that sustains this motion has magnitude
F = mv^{2}/r.
This follows from applying F=ma to equation (3). We saw in the derivation of that equation that the direction of acceleration was radially inward; this is now supported by our intuition that the force is toward the center of the circle.
Torque
Although a force could be acting in any direction, it is only the component of the force in the direction perpendicular to the motion that sustains circular motion. Thus we can say that a centripetal force is the perpendicular component of a force. Now let’s consider the effect of the rest of the force – i.e. the tangential component of the force. (See Figure 2-7(a).) It is an obvious trigonometric fact that the magnitude of this component is Fsinθ , where θ is the angle between the force and the radial line. This tangential component is the component that causes changes to the circular motion.
Figure 2-7. (a) The tangential component of the force is F sin θ . (b) The length of the radial vector times the tangential component of the force is the same as the length of the force times the perpendicular component of the radial vector.
You have probably noticed that when you try to push open a door, the closer you push to the side of the door where the knob is located, the easier it is to accomplish opening the door. In other words, the further your push is from the hinges, i.e. from the axis of the door’s rotation, the more effective your force is at causing the door to rotate. That, of course, is why the knob is located on that side of the door. It is also why cat doors should be located on the other side – so that if the human door happens to be slightly ajar when you go through the cat door, it won’t disturb you by moving if you happen to brush against a side of the cat door.
If you’re extremely observant as well as experimentally inclined, you may even have noticed that when you push twice as far from the axis of rotation, you only have to push half as hard to cause the same rotational motion of the door. Regardless, the observation that the effect of the tangential force on rotation varies directly with the distance from the axis of rotation hopefully leads to the conjecture that the dependency is linear, i.e.. that the effect is proportional to the distance from the axis to the point where the force is applied. We’ll call this radial distance r. It’s the magnitude of the radial position vector, which points from the axis of rotation to the point where the force is being applied.
Thus we have the conjecture that the rotation depends on the quantity
(4)
r F sinθ.
Indeed it does, and this quantity is actually the magnitude of a vector called the torque, or the moment of the force.
Torque is symbolized by the Greek letter τ (tau). In SI units, it is measured in newton-meters (Nm). Although these are the same units as those of joules, which measure work and energy, torque is not a form of energy. Note that the distance being considered here is perpendicular to the direction of motion, whereas for work the distance is along the direction of motion.
Torque influences rotational motion analogous to the way force influences translational motion. In the absence of torque, there is no change to rotational motion. Note that the absence of torque does not mean the absence of force: A force being applied at the center does not constitute torque because r = 0 for it; and a force being applied parallel to the motion arm does not constitute torque because sin θ = 0 for it (in fact, that’s the centripetal force that is sustaining the rotation).
In translational mechanics, the natural state of motion is one of constant velocity, which may be zero at rest, and any changes to it require a force. In rotational mechanics, the natural state of motion is rotation at a constant angular velocity, and changes to it require a torque.
The amount of torque is the radial distance r times the tangential component of the force, the component of the force perpendicular to the radial vector. Another way to view it is as the magnitude of the force times the component of the radial vector perpendicular to the line of action of the force. (See Figure 2-7(b).)) This distance that is the perpendicular component of the radial vector is called the moment arm or the lever arm. The latter term is due to the fact that simple lever motion is based on this principle. Thus the magnitude of the torque is the magnitude of the force times the moment arm. Using both momentous terms, we can say the moment of the force is the product of the moment arm and the force.
]]>https://fizyxforfelines.wordpress.com/2010/03/26/chapter-2-excerpt-2/feed/0skonabrittainChapter 2 – Excerpt 1
https://fizyxforfelines.wordpress.com/2010/03/19/290/
https://fizyxforfelines.wordpress.com/2010/03/19/290/#respondFri, 19 Mar 2010 18:25:31 +0000http://fizyxforfelines.wordpress.com/?p=290In Chapter 2, we are no longer restricted to motion in a straight line, but take into account all that twisting and turning that can get you out of tight places. This first excerpt includes the French mathematician and philosopher René Descartes, famous for connecting algebra to geometry and infamous for throwing a cat out of a window.
If your head is spinning after getting through Chapter 1, you’ve come to exactly the right place – Chapter 2. This chapter is all about such spinning motion!
So far we have been discussing translational motion – motion in a straight line in a particular direction. Although we have considered curved trajectories, such as the parabolic arc of a tossed ball or a pouncing cat, we have done so by analyzing the separate translational components of the motion in the horizontal and vertical directions. And even then the motion was along a line, even if the line wasn’t straight. For such translational motion, we can treat an extended body as if all its mass were concentrated at a point – the point called its center of mass – and all the forces were acting at that point. But this simplification misses local features of the motion, motion of the object with respect to itself, such as rotational spin.
In order to capture those features in our description, we need to learn about the rotational analogs of all the translational motion quantities we have considered. And we will see that the corresponding principles also hold true. Some of these analogs, along with their symbols, are presented in the table in Figure 2-1. The table is mainly for later reference – don’t worry if at this point it’s all Greek to you.
Figure 2-1.
[Linear] Displacement (s)
Angular Displacement (θ)
[Linear] Velocity (v)
Angular Velocity (ω)
[Linear] Acceleration (a)
Angular Acceleration (α)
Force (F)
Torque (τ )
Mass (m)
Rotational Inertia (I)
Newton’s 2nd Law: F = m a
τ = Iα
Rotational motion is circular motion around a line called the axis of rotation. If it is an internal axis, one that goes through the rotating body, then the motion is called rotation; and if it’s an external axis, it’s called revolution. Thus we say that the earth rotates around the axis through its geographical poles, whereas it revolves around the sun, or rather around a line through the sun perpendicular to the plane of the earth’s orbit. (The earth’s orbit is actually an ellipse but it’s of such low eccentricity – 0.0017 – that it’s essentially a circle. Even the lunar orbit, with an eccentricity of 0.055, is practically circular.)
Q. Is Exeter, our exercising exercise icon, rotating or revolving? (Perhaps as a compromise, we could say he’s “revolting”.)
Linguistic Note
Note that the term “rotational motion” subsumes revolving as well as rotating. This may be because “revolutional” is not a word and the adjective “revolutionary” has taken on another meaning. Accordingly, we will sometimes use the word “rotation” generically instead of always repeating the more cumbersome phrase “rotation or revolution”. And we will always use the more common term “axis of rotation” even when the rarely used “axis of revolution” would be more correct.
However, that other meaning of “revolution” actually has its roots in rotational physics! About half a millennium ago, Nicolas Copernicus (1473-1543), a Polish astronomer, presented the startling idea that the planets revolve around the sun, in a work entitled De revolutionibus orbium coelestium (On the Revolutions of the Celestial Spheres). This heliocentric theory had a radical effect on the geocentric European scientific community’s worldview. (In fact, even in the next century, Galileo was arrested for supporting it.) Not only was Copernicus’s work the genesis of modern astronomy, but its title word “revolution” took on the meaning of "an action leading to a radical change in society".
For rotational motion, it is often more convenient to use polar coordinates than rectangular Cartesian coordinates. We assume you are familiar with polar coordinates from studying trigonometry, and present Figure 2-2(a) just for review and reference.
Conversion Equations:
x = r cos θ r^{2} = x^{2} + y^{2} y = r sin θ tanθ = y/x
Figure 2-2. (a) The point P has rectangular coordinates (x,y) and polar coordinates (r,θ). (b) The arc length is simply rθ when θ is measured in radians, because it is a portion equal to θ/2π of the whole circumference, which is 2πr.
Temporarily abandoning the Cartesian coordinate system, which was named after the 17^{th} century French mathematician and philosopher René Descartes, should be especially pleasing to those of you who have heard of Descartes’ feline defamation activities. Not only did he frequently proclaim that “cats have no souls”, but he once even attempted to demonstrate it by throwing his cat out of an upper-story window^{1}. Perhaps the roots of Descartes’ anti-feline tendencies were first exhibited when he was an advanced teenager obtaining his law degree at the University of Poitiers. As pointed out by physicist Stephen Hawking of Cambridge University, author of the bestselling physics book of all time, Descartes was more interested in studying medicine at Poitiers and while there “he developed a keen interest in dissection.”^{2}
Angles are typically measured in either degrees (symbolized °) or radians (abbreviated rad). A full circle has 360 degrees or 2π radians. Since the circumference of a circle is proportional to its radius, with the constant of proportionality being 2π, when we use radians to measure angles, we have the very nice property that arc length is simply radius times angle, or, in symbols,
(1)
s = rθ
(see Figure 2-2(b)). Note that a radian is a dimensionless quantity – there are no labels, in any system of units, to a measurement in radians.
Thus we take our basic unit of rotational motion to be angular displacement measured in radians. If the motion is counterclockwise, the angular displacement is considered positive; clockwise motion has a negative angular displacement.
For a rigid body undergoing rotational motion, all parts always have the same angular displacement. For example, in Figure 2-3, borrowed from Paul Hewitt’s Conceptual Physics textbook, the cat has apparently been scared into rigidity by being placed on a rotating turntable. Thus all its parts are simultaneously undergoing the same angular displacement.
Figure 2-3. A revolving rigid body.
Angular speed is a measure of the angle traversed per unit time. More precisely, angular speed is the derivative of angular displacement with respect to time. It is symbolized by the lowercase Greek letter omega – ω. As an advantage of using radians, we also obtain a nice simple relationship between the angular speed ω and the associated linear speed v, by differentiating equation (1): v = ds/dt = d(rθ)/dt = r dθ/dt = rω. so we have
(2)
v = rω
Since angular displacement is dimensionless, the units of angular speed are merely inverse seconds.
Another popular way to measure rotational speed is by counting the number of rotations or revolutions per unit of time. Historically, that unit of time was often taken to be a minute, giving rise to a rotational speed unit of revolutions per minute, commonly abbreviated RPMs. As some of you might happen to know from a former lifetime, the rotational speed of the cat in Figure 2-3 is either 33 1/3 RPM or 45 RPM.
The quantity of revolutions per second is called frequency and symbolized f. When we use seconds as the unit of time, we speak of cycles per second, or cps, also called hertz, abbreviated Hz, after the 19^{th} century German physicist Heinrich Hertz. (Hertz discovered radio waves, which have a frequency ranging from about ten kHz to about a hundred GHz.)
To distinguish these two ways of measuring speed, we call the number of rotations or revolutions per second the rotational speed or frequency f, and the number of radians per second the angular speed or frequency ω. They are, of course, simply related by ω = 2πf.
The reciprocal of frequency, the amount of time it takes for one complete rotation or revolution, is called the period of the motion, and symbolized T, presumably standing for time. Thus, T = 1 / f = 2π / ω.
Q. What is the frequency of the earth’s rotation around its axis, in both hertz and rad/s? A. Since the period is known to be one day, to get the frequency in hertz, we just need to convert inverse days to inverse seconds.
Q. What is the frequency of the revolution of the mouse at the end of the second hand in the popular vintage wristwatch shown in Figure 2-4?
Figure 2-4. Once a minute, at about 40 seconds past the minute, the cat on the watch face appears to be reaching toward the mouse on the second hand.
For a rigid body, the angular speed is the same at every point, while the linear speed is proportional to r, the distance from the axis of rotation. Again referring to Figure 2-3, the left forepaw is about twice as far from the center of the turntable as the right back paw. So while they both have the same rotational speed, the linear speed of the left forepaw is twice as great as the linear speed of the right back paw. In circular motion, the linear speed v is also known as tangential speed, because the instantaneous direction of linear motion is tangent to the circle.
1 – According to Robert G. Brown, a colorful physics professor at Duke University, the fate of that cat is not known. source: http://www.phy.duke.edu/~rgb/Philosophy/axioms/axioms.pdf. <return-to-text>
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https://fizyxforfelines.wordpress.com/2010/03/19/290/feed/0skonabrittainChapter 1 – Excerpt 6
https://fizyxforfelines.wordpress.com/2010/03/12/chapter-1-excerpt-6/
https://fizyxforfelines.wordpress.com/2010/03/12/chapter-1-excerpt-6/#respondFri, 12 Mar 2010 20:03:59 +0000http://fizyxforfelines.wordpress.com/?p=279This is the last excerpt from Chapter 1 and also the shortest one.
Energy
Another quantity dependent on mass and velocity is ½ m|v|^{2}, which is called kinetic energy. Note that it is a nonnegative scalar quantity.
Kinetic energy is just one form of the most familiar and fundamental physical quantity, namely energy. Although the concept of energy is a commonplace one, it is rather difficult to define. Essentially, energy is the capacity to effect motion with respect to a force.^{11}
Don’t despair if you don’t fully grasp the concept of energy from this definition, any more than we despair that we haven’t provided you with a better definition. About a century ago, the famous French mathematician Henri Poincaré (who turned to writing popular science when he got too old to do mathematics, and was so good at it that he was eulogized as “a kind of bard of science”) stated “we cannot give a general definition of energy”. According to some modern textbooks:”there is no completely satisfactory definition of energy”^{12}; "we really don’t know what energy is but we know the many forms it takes"^{12.1}; "a kind of ‘quick-change artist’ with a whole trunkful of disguises"^{12.2}. And our favorite Nobel Laureate, Richard Feynman, also a brilliant explicator, went further: “It is important to realize that in physics today, we have no knowledge of what energy is”^{13}.
The SI unit of energy is the joule, which is named after James P. Joule and abbreviated J. A joule is a newton-meter. Thus, one joule is the amount of energy it takes to lift a small apple (about .1 kg) up one meter from the ground. On the other hand, the food energy your person obtains by eating such an apple is 50 Calories, or about 200 kilojoules, since one Calorie is exactly 4184 joules by definition. Unlike Newton, Joule did not have any cats and did have a wife. But that is just as well, since he was infamous for meeting with Lord Kelvin to conduct one of his famous experiments while he was a newlywed on his honeymoon, so one can only imagine how he would have neglected a pet cat.
Here we will just focus on the forms of energy relevant to classical mechanics, i.e. mechanical energy, which takes two forms – kinetic energy and potential energy. Kinetic energy and potential energy are often obscurely symbolized by T and U, respectively, but we will instead use the more obvious K.E. and P.E.
Kinetic energy is due to motion. From the above definition of K.E. = ½ mv^{2}, it is clear that only moving objects have kinetic energy, and the faster they’re moving, the more kinetic energy they have.
In contrast, potential energy is due to position. It is stored energy that could be released by a change of position. For example, a compressed or stretched spring or elastic string has a potential energy that depends on how far it is from the relaxed position. The elastic potential energy of a spring either compressed or expanded a distance x is P.E. = ½ kx^{2}, where k is the spring constant (i.e. the spring’s force is F = -kx). Gravitational potential energy is possessed by elevated objects with respect to a lower position, for they have the potential to move to that lower position. For an object elevated at height h, the gravitational potential energy is P.E. = weight×height = mgh^{14}. Potential energy is illustrated in Figures 1-12, 1-13 and 1-14.
Figures 1-12 & 1-13 & 1-14 & 1-15. There’s elastic potential energy in stretched or scrunched elastic strings, springs, and muscles of ready-to-pounce crouched animals. And there’s additional gravitational potential energy when the pounce takes place from above.
Kinetic energy and gravitational potential energy are clearly interchangeable. Consider an object at rest at height h. As it falls, its motion is described by Equations (2) with the initial speed v_{0} = 0 and the initial height z_{0} = h:
v = – gt
z = -½gt^{2} + h = h – ½gt^{2}
At any point t in time, its kinetic energy is
K.E. = ½ mv^{2} = ½ m(-gt)^{2} = ½ mg^{2}t^{2}
and its potential energy is
P.E. = mg(h – ½gt^{2}) = mgh – ½ mg^{2}t^{2}
So its total mechanical energy is
M.E. = K.E. + P.E. = ½ mg^{2}t^{2} + (mgh – ½ mg^{2}t^{2}) = mgh
which is independent of the time t. Thus, as it falls, all its potential energy is gradually transformed into kinetic energy, with the total always being constant, equal to the initial potential energy it had when it had no kinetic energy because it was still.
Energy considerations also give us a very easy way to calculate the speed of the object when it hits the ground, without knowing what the time t is then. At that point, all the energy is kinetic, so we just solve ½ mv^{2} = mgh to get v = (2gh)^{½}. Note that although we didn’t need to know the landing time to obtain this result, we can now use it to easily determine that time: t= (2h/g)^{½}. Not only did we avoid calculus here, but we didn’t even have to solve a non-trivial quadratic equation, as we did when analyzing motion in the first section of this chapter. This is, in a sense, the opposite approach. Energy considerations tend to lead to simpler formulations of physical laws.
It is not just the two forms of mechanical energy that are interchangeable. More generally, all types of energy can be interchanged. For example, when the falling object just discussed reaches the ground and stops moving, it has neither kinetic energy nor gravitational potential energy. But its energy was transferred to the spot of ground it hit, which heated up.
Energy takes many forms: heat, light, chemical, mechanical, electrical, radiant, atomic, nuclear, etc. As Eugene Hecht said, “we bake apple pies with thermal energy and defend them with nuclear energy“^{15}. The transformations between the different forms of energy account for all the physical phenomena that we observe, such as current, combustion, and color changes, as well as motion.
Although energy can be transformed between its different types, energy cannot be either created or destroyed – the total amount of energy in the universe is a constant. This is the greatest of the conservation laws in physics, and one of the greatest of all generalizations in physics. It is known as the Principle of the Conservation of Energy. At various times throughout history, it has appeared that it was being violated, i.e. that energy was being created or destroyed, but further investigation of the suspect phenomena instead led to the discovery of new forms of energy, and occasionally even to new laws of physics.
Unlike virtually all of the other concepts in this chapter, the idea of energy as a physical concept was not only not due to either Newton or his cat, but it was not even known by either of them.
There are many additional concepts of linear classical mechanics, such as elasticity, impulse and power, as well as work, which have been omitted from this chapter in the interests of saving space and patience. If you wish to pursue them, we suggest you start by pouncing on a mouse connected to the Internet.
11 – The laborious nature of this definition is due to the circumventing of the definition of the physical concept of work – thus we are doing work to avoid “work” here! (Technically, energy is the capacity to do work, where work is defined as force x distance, for the motion through a distance of an object due to a force, or W = ∫Fdx.) Linguistically, the root of “energy” is the Greek ergon, which means work. <return-to-text>
14 – The elastic potential energy is the amount of work it takes to compress or expand a spring that distance, and the gravitational potential energy is the amount of work it takes to lift the object to that height. <return-to-text>