Numbers in Physics PHYSICS 107 Lecture 3 Numbers and Units We've seen already that even 2500 years ago Aristotle recognized that lengths and times are magnitudes, meaning that any length or time can be described by a number. He also recognized that speed s was a combination of length and time, and that if you double the length you double the speed, but that if you double the time you halve the speed. This really suggests that we should write that the speed is equal to distance traveled divided by time elapsed: s = d / t, since this equation automatically gives those mathematical properties to the concept of speed. However Aristotle did not take that step (I believe) because it would mean dividing incommensurate magnitudes by one another. He would have regarded our equation as an apples vs. oranges type of mistake. You can divide a length by a length: 2 miles divided by 1 mile should just be 2, for example. But to him the length divided by the time doesn't make sense. What I'm going to suggest here is that we just write s = d / t but when we actually divide things we only divide lengths by lengths and times by times, and similarly for other quantities. This will allow us to think of miles per hour as being a speed and we can divide a speed by a speed, they being like magnitudes. Now let s think about how this would work. The most common way to describe speed in everyday life for us is in miles per hour. Thus, 3 mph is walking speed, 15 mph is bike-riding speed, 30 mph is car-in-town speed, 60 mph is car-on-thehighway speed, 200 mph is fast train speed (though not in the US), and 500 mph is airplane cruising speed. Now we can answer some questions like: how much faster is a cruising plane than a walking person? The answer is 500 / 3 = 167. That's a ratio of speeds so we only divided a like by a like: a speed by a speed. Even Aristotle would've been okay with that. And actually this ratio can tell us some useful things. We know that it takes about two hours to fly from Madison to New
York. How long would it take us to walk from Madison to New York? Well the answer is 167 times as long, which is 2 167 hours or 334 hours. If you walk about 10 hours a day, it would take you about a month to get to New York. So it's useful to use numbers to describe lengths, times AND speeds. But we do need to be careful to always only divide like by likes to get an actual number. Units I ask someone how far it is from here to Chicago. They answer me 150. I ask someone else the same question. They answer me 240. Who is right? Well, both are right and both are wrong, in a sense. 240 km is a right answer, and 150 miles is a right answer. So the original answers are not quite wrong, but they're definitely incomplete. You must give the unit of length in order to give a really right answer. An even better answer in our society, in which you probably assume you ll be driving, is three hours. Why? Well s = d / t and so t = d / s, according to the usual rule of algebra regarding cross-multiplication. So t = 150 miles divided by 50 mph. Here 50 mph is roughly the average speed you'll be able to go on the way from Madison to Chicago in a car. (This is additional information I didn t get when somebody told me 150 miles). Thus t = 150 / 30 h. Notice that in this calculation we only ever divided miles by miles, that is we divided a like unit by a like unit and we used the rule from algebra x = 1 / (1/x). Again we see that the idea of defining the speed to be the distance divided by the time is surprisingly useful. We only ever divided likes by likes, the same unit divided by the same unit, and we used only basic rules of algebra and we got to a useful answer. This kind of manipulation of numbers will come up over and over again in the course. So it's useful for you to get used to it and we will assign some problems so that you can get some practice doing it. Now let's go back to the 240 versus 150 question. Kilometers and miles are both measures of length. So they can be divided by one another. And because they are fixed magnitudes, the ratio is fixed: so we have that 1 mile divided by 1 km is equal to 1.61, which means that a mile is longer than a kilometer. In symbols 1 mi / 1 km = 1.61. Also 1 km / 1 mi = 1/1.61 = 0.62. 0.62 is the also the ratio of 150 to 240, as you can easily see by just doing the math.
Any length unit can be divided by any other length unit and you get just a number. Let's see how this works. Let's ask the slightly silly question: how many feet is it to Chicago? The answer is 150 miles, as we already said. Now I am going to multiply this number by one, which does not change its magnitude. But I'm going to express the number one in a strange way: 1 = 5280 ft / 1 mi, so the number of feet to Chicago is 50 mi (5280 ft / 1 mi) = 150 x 5280 ft which is 792,000 ft. How many meters is it to Chicago? Well that's 240 km times 1000 m divided by 1 km equals 240,000m. You can see that the calculation in SI units (meters, grams, seconds) is simpler than the one in English (feet, pounds, seconds) units, which is why most of the world has converted to SI units. We can do other kinds of manipulations of units, always respecting the rule that we can only divide like magnitudes by like magnitudes to get pure numbers, and following the rules of algebra. So, for example, how many square feet in a square yard? This is one which does come up in practical problems like figuring out the area of floor space. The answer is pretty simple: (1 yd) 2 = 1 yd (3 ft / 1yd) 1 yd (3 ft / 1yd) = 9 (ft) 2. Scientific Notation Let's return to speed or velocity. We have defined it as distance divided by time: s = d/t. One speed we will be talking about a lot in this course is the speed of light. It is always denoted by a small c. Its value can be measured very accurately and it is c = 299,792,458 m/s. Within a fraction of a percent, this is equal to 300,000,000 m/s. It's obviously silly to write all the zeros. It will get worse because we'll have even bigger numbers to deal with later on, and the way to avoid lots of unnecessary writing is to use scientific notation. We notice that 100 is equal to 10 10 which is equal to 10 2. 1000 is equal to 10 10 10 = 10 3. The rule is that the exponent is just equal to the number of zeros. Notice that 10,000 = 100 100 = 10 2 10 2 = 10 4. This gives the rule for multiplying numbers in scientific notation: we add the exponents. So c is equal to 3x100,000,000 is equal to 3 x 10 8 m/s. We can also write the accurate value for c in scientific notation: c = 2.99792458 x 10 8 m/s.
Scientific notation is particularly good when the numbers get really big. Another unit of length that we are going to be talking about quite a bit later in this course is a light year, abbreviated as lyr. It's the distance that light travels in one year which is equal to the speed of light times one year and one year is 3.14 10 7 s so 1 lyr = (3 10 8 m/s) 3.14 x 10 7 s = 6.26 10 15 m. In kilometers this would be 9.42 10 15 m (1 km / 1000 m) which is equal to 9.42 10 1 2 km. Again all these manipulations are easier in SI units. A light year is very roughly equal to the distance between nearby stars. Alpha Centauri, the star nearest to the sun is 4.4 lyr away from the earth. So if you look in the sky and see the star, the light is already 4.4 years old. Scientific notation is also very good for talking about very small numbers. We'll be talking a lot about atoms in this course, for example. They are very small. The radius of a hydrogen atom is about 5.29 x10-11 m which is 0.0000000000529 m. Again the exponent is the number of zeros, but this time the exponent is negative. How long does it take light to cross an atom? The distance is twice the radius, so we start with s = d / t and from there algebra tells us that t = d / s, so t t = 2 5.29 x 10-11 m / (3 10 8 m / s) = 3.52 (10-11 / 10 8 )s = 3.52 10-11-8 s = 3.52 x 10-19 s. The international system provides a number of standard abbreviations for lengths and times. In decreasing order we have 1 km = 10 3 m, 1 km = 10 3 m, 1 m = 10 0 m 1 mm = 10-3 m, 1 μm = 10-6 m, 1 nm = 10-9 m, 1 pm = 10-12 m, 1 fm = 10-15 m. In words these symbols are for kilometers, meters, millimeters, micrometers, nanometers, picometers, and femtometers. The system is the same for seconds, except that I don't know that anyone talks about a kilosecond very much. When dealing with a given subject, there are usual natural choices of units just for convenience. So in astronomy we ll often use light years, while when one is talking about atoms, for example, to use nanometers instead writing 10-9 m all the time. Significant digits You'll probably have noticed that I did not worry too much about making very precise calculations in what we have done so far that's because we have been
mostly interested in powers of 10, which only involves relatively rough calculations. I mostly used only two or three nonzero digits in every number. Why? This actually brings up an extremely important point: in physics, we are not interested in numbers for their own sake! We are only interested in understanding nature. Numbers are just a tool for understanding. So in general when we re discussing some topic in physics, we only use the figures that are significant to understand the particular subject matter. In the case of discussion of scientific notation, using too many significant figures would have distracted from the points I was trying to make, so I only used two or three in most cases. In fact two or three significant figures is usually enough for almost any physics discussion and sometimes even one significant figure is sufficient. There are exceptions. Sometimes a number can be measured to very high accuracy and a theory for that number can also make an extremely accurate prediction. In such a case it makes sense to use a larger number of significant figures since we would like to test the prediction as well as we can. Later, we ll see that there is a particle called the muon and it has a magnetic moment that can be measured to 1 part in 100 million, and the theory for it also accurate to about the same level. What s more, they agree!