Chapter 1 – Excerpt 1
This post is the beginning of the first “real” chapter, which is about linear motion. After a historical note about Newton and his cat, some vector concepts are presented, with a pedagogical use of color, to set the framework for the study of motion. (Despite their Zen-like propensity for stillness, cats are known to occasionally engage in motion.)
Fizyx For Felines
Chapter 1 – Newton’s Cat
Classical Mechanics – A Straightforward Approach
~ Historical Note ~
Sir Isaac Newton (1642-1727) was one of the greatest physicists of all time, although historians still debate how much of this greatness should be attributed to his cat.
He was born the year that Galileo died, in Woolsthorpe, Lincolnshire, a small village about 100 miles north of London. As a youngster, he was too bright for the village school, so he was sent eight miles away to grammar school in Grantham, where he lodged in the apothecary’s house. He soon became the top student at the new school, as well as the fiancé of the apothecary’s stepdaughter, Anne Storer. But then, at age 19, he left for Cambridge University, where he became so distracted by his studies that she married someone else. There is evidence that Newton used boards in both the apothecary garret and the school library in Grantham as scratching posts – carvings of his name remain in both places1.
Historians believe that Newton never had another romantic relationship, and that he died a virgin. It is suspected that he had Asperger’s Syndrome, which is a form of autism. Some of you readers may be familiar with this condition, since current medical practice encourages pets for autistic children.
Not to dispute the historians’ theory that Newton died a virgin, but one must not confuse this lack of sex with a lack of either romance or physical affection in his life. Of course his main sweetheart was a cat, whom he called Spitface2, and who may have been Galileo’s last cat in her previous life.
In addition to being a great physicist, Isaac Newton was a great mathematician. In order to describe his laws of motion, which he had developed by age 24, he invented a whole new branch of mathematics – the calculus.
Newton’s mathematical prowess extended not only to the felines in his family but even, to a limited extent, to the canines. According to the diary of his friend John Wallis (the English mathematician who introduced ∞ as a symbol for infinity), one day Newton claimed to Wallis “My dog Diamond knows some mathematics. Today he proved two theorems before lunch”. But when Wallis exclaimed “Your dog must be a genius!”, Newton replied “Oh, I wouldn’t go that far. The first theorem had an error and the second had a pathological exception”.4
It is a well-established historical fact that Newton’s cats never created any proofs with errors in them.
T he basic laws of classical mechanics are the Laws of Motion known as Newton’s Laws. Even if you skipped the historical note above, you’ve probably heard of Sir Isaac Newton, a 17th century English physicist. Some of you more mature readers may even have met him in one of your previous lives, as he was known to be quite fond of cats. In order to develop and use these laws, we need to introduce some mathematics, as Newton himself did.
Although these few pages of mathematical background material may seem rather dry, if you get through them, you will be rewarded later in this chapter with an application that has possibly life-saving ramifications, which depends on your understanding of the concept of a vector.
Many quantities that physicists use cannot be completely encapsulated by pure numbers. For example, to describe the motion of an object, we would not only say how fast it is going, but also in what direction it is moving. The mathematical object used to represent quantities that have both size and direction is a vector. By contrast, directionless quantities that are represented by pure numbers, such as temperature, are called scalars. In order to distinguish vectors and scalars in this text, we will always use the convention of non-italic boldface type for vectors.
We symbolize vectors with arrows. See Figure 1-1. The direction in which the arrow is pointing indicates the direction of the associated quantity, and the length of the arrow corresponds to the vector’s size, which is called its magnitude. Note that magnitude is symbolized with the same vertical bars as used for absolute value, since absolute value is just the one-dimensional analog of vector magnitude. Vectors with magnitude 1 are called unit vectors, and i, j, and k are the conventional names for the unit vectors in the x, y and z directions, respectively. The projection of a vector in any of these directions is called its component in that direction, and the magnitude of a vector is calculated from all its components by the Pythagorean Theorem4.5.
Often, a set of three scalar equations, one for each direction in our three-dimensional world, can be replaced by one vector equation, which looks just as simple as any one of the scalar ones. This is an example of how higher-powered mathematics allows more concise formulations of physical laws.
We usually use a coordinate system in which the x- and y-axes are on the ground (or a table, refrigerator top or other level surface) and the z-axis points straight up. Of course, the laws of physics are independent of the coordinate system – because physical reality doesn’t depend on our conceptualization of it – but it’s convenient to express them that way.
There are a few basic, related quantities that are used to describe the motion of an object: its position, its velocity, and its acceleration. Velocity is the derivative of position with respect to time and acceleration is the derivative of velocity with respect to time; therefore, acceleration is the second derivative of position with respect to time. The SI units of these three quantities are thus m, m/s and m/s2, respectively, where m stands for meters and s stands for seconds.
An object’s position is indicated by its displacement from a point called the origin. (In fact, the words "position" and "displacement" are used synonymously for this quantity in classical mechanics.) Obviously this position must be specified by a vector, since we don’t live in a semi-finite one-dimensional world. For example, if we take the origin to be on the surface of the dinner table, then if you leap up onto the top of the refrigerator, the z-coordinate of your position will be positive, whereas if you jump down to the floor, it will be negative. Distance, a nonnegative quantity, is the magnitude of the displacement vector.
Q. Suppose you climb straight up a 2-meter fence and stroll along the top for 3 meters. What is your net displacement?
A. Your position is the vector sum as illustrated in Figure 1-2. By the Pythagorean Theorem, its magnitude is sqrt(32+22) = sqrt(13) ≈ 3.6 meters. Its direction is tan-1(3/2) ≈ 56.3° from the vertical. In navigation applications, it is more common to refer instead to the angle of elevation or depression, which is the angle θ between the line of sight and the horizontal. That angle is about 33.7° by the Complementary Angle Theorem, an elementary trigonometry result with a particularly euphonic acronym.
2 – http://www.catworld.co.uk/articlecatworld.asp?artid=923&pre=25638 em>What’s in a name? article by Dominique Lummus in issue 282 of Cat World <return-to-text>
3 – The position occupied since 1979 by Stephen Hawking, who was born 300 years later than Newton.<return-to-text>
3.5 – Not his former fiancée.<return-to-text>
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