Chapter 2 – Excerpt 1
In 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.
Fizyx For Felines
Chapter 2 – Newton’s Cat’s Kittens
Classical Mechanics – From Another Angle
It’s all Greek to me.
from a Medieval Latin proverb0
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.
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”.)
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.
Temporarily abandoning the Cartesian coordinate system, which was named after the 17th 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 window1. 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,
(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.
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
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 19th 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?
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?
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.
2 – Hawking, Stephen, God Created the Integers: The Mathematical Breakthroughs that Changed History, page 286, Running Press ©2005. <return-to-text>
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