## 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.

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Chapter 2 – Newton’s Cat’s Kittens Classical Mechanics – From Another Angle It’s all Greek to me. from a Medieval Latin proverb
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 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 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 For rotational motion, it is often more convenient to use
Temporarily abandoning the Cartesian coordinate system, which was named after the 17 Angles are typically measured in either degrees (symbolized °) or
(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
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 19 To distinguish these two ways of measuring speed, we call the number of rotations or revolutions per second the 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 0 – Source: World Wide Words by Michael Quinion, http://www.worldwidewords.org/qa/qa-gre1.htm <return-to-text> 1 – According to 2 – Hawking, Stephen,
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