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For Dummies is a registered trademark of Wiley Publishing, Inc. in the United States and other countries. Used here by license.Physicists have a way of getting their minds into the darndest places, and those places often involve really big or really small numbers. For example, say you’re dealing with the distance between the sun and Pluto, which is 5,890,000,000,000 meters. You have a lot of meters on your hands, accompanied by a lot of zeroes. Physics has a way of dealing with very large and very small numbers; to help reduce clutter and make them easier to digest, it uses scientific notation. In scientific notation, you express zeroes as a power of ten — to get the right power of ten, you count up all the places in front of the decimal point, from right to left, up to the place just to the right of the first digit (you don’t include the first digit because you leave it in front of the decimal point in the result). So you can write the distance between the sun and Pluto as follows: 5,890,000,000,000 meters = 5.89 × 1012 meters Scientific notation also works for very small numbers, such as the one that follows, where the power of ten is negative. You count the number of places, moving left to right, from the decimal point to just after the first nonzero digit (again leaving the result with just one digit in front of the decimal): 0.0000000000000000005339 meters = 5.339 × 10-19 meters If the number you’re working with is greater than ten, you’ll have a positive exponent in scientific notation; if it’s less than one, you’ll have a negative exponent. As you can see, handling super large or super small numbers with scientific notation is easier than writing them all out, which is why calculators come with this kind of functionality already built in.
Checking the Precision of MeasurementsPrecision is all-important when it comes to making (and analyzing) measurements in physics. You can’t imply that your measurement is more precise than you know it to be by adding too many significant digits, and you have to account for the possibility of error in your measurement system by adding a ± when necessary. The following sections delve deeper into the topics of significant digits and accuracy.
Knowing which digits are significantIn a measurement, significant digits are those that were actually measured. So, for example, if someone tells you that a rocket traveled 10.0 meters in 7.00 seconds, the person is telling you that the measurements are known to three significant digits (the number of digits in both of the measurements). If you want to find the rocket’s speed, you can whip out a calculator and divide 10.0 by 7.00 to come up with 1.428571429 meters per second, which looks like a very precise measurement indeed. But the result is too precise — if you know your measurements to only three significant digits, you can’t say you know the answer to ten significant digits. Claiming as such would be like taking a meter stick, reading down to the nearest millimeter, and then writing down an answer to the nearest ten-millionth of a millimeter. In the case of the rocket, you have only three significant digits to work with, so the best you can say is that the rocket is traveling at 1.43 meters per second, which is 1.428571429 rounded up to two decimal places. If you include any more digits, you claim an accuracy that you don’t really have and haven’t measured. When you round a number, look at the digit to the right of the place you’re rounding to. If that right-hand digit is 5 or greater, you should round up. If it’s 4 or less, round down. For example, you should round 1.428 to 1.43 and 1.42 down to 1.4. What if a passerby told you, however, that the rocket traveled 10.0 meters in 7.0 seconds? One value has three significant digits, and the other has only two. The rules for determining the number of significant digits when you have two different numbers are as follows:
When you multiply or divide numbers, the result has the same number of significant digits as the original number that has the fewest significant digits. In the case of the rocket, where you need to divide, the result should have only two significant digits — so the correct answer is 1.4 meters per second.
When you add or subtract numbers, line up the decimal points; the last significant digit in the result corresponds to the right-most column where all numbers still have significant digits. If you have to add 3.6, 14, and 6.33, you’d write the answer to the nearest whole number — the 14 has no significant digits after the decimal place, so the answer shouldn’t, either. To preserve significant digits, you should round the answer up to 24. You can see what I mean by taking a look for yourself:
3.6
+14
+ 6.33
23.93
By convention, zeroes used simply to fill out values down to (or up to) the decimal point aren’t considered significant. For example, the number 3,600 has only two significant digits by default. If you actually measure the value to be 3,600, of course, you’d express it as 3,600.0, with a decimal point; the final decimal point indicates that you mean all the digits are significant.
Estimating accuracy
Physicists don’t always rely on significant digits when recording measurements.
Sometimes, you see measurements such as
5.36 ± 0.05 meters
The ± part (0.05 meters in the preceding example) is the physicist’s estimate of the possible error in the measurement, so the physicist is saying that the actual value is between 5.36 + 0.05 (that is, 5.41) meters and 5.36 – 0.05 (that is, 5.31 meters), inclusive. (It isn’t the amount your measurement differs from the “right” answer as given in books; it’s an indication of how precise your apparatus can measure — in other words, how reliable your results are as a measurement.)
Fathoming the ± fad
This ± business has become so popular that you see it all over the place now, as in a real-estate ad that announces 35± acres for sale. Sometimes, you even see real-estate ads with numbers such as ±35 acres, which makes you wonder whether the agent realizes that the ad means the actual acreage is in the range of –35 to +35 acres. What if you buy the place and it turns out to be –15 acres? Do you owe the agent 15 acres?
Arming Yourself with Basic Algebra
Yep, physics deals with plenty of equations, and to be able to handle them, you should know how to move the items in them around. Time to travel back to basic algebra for a quick refresher. The following equation tells you the distance, s, that an object travels if it starts from rest and accelerates at a for a time t: s = 1⁄2at2 Now suppose the problem actually tells you the time the object was in motion and the distance it traveled and asks you to calculate the object’s acceleration. By rearranging the equation, you can solve for the acceleration:
a = 2s / t2
In this case, you multiply both sides by 2 and divide both sides by t2 in order to isolate the acceleration, a, on one side of the equation. What if you have to solve for the time, t? By moving the number and variables around, you get the following equation: t a 2s = Do you need to memorize all three of these variations on the same equation? Certainly not. You just memorize one equation that relates these three items — distance, acceleration, and time — and then rearrange the equation as needed.
Tackling a Little Trig
Besides some basic algebra, you need to know a little trigonometry, including sines, cosines, and tangents, for physics problems. To find these values, you start out with a simple right triangle — take a look at Figure 2-1, which displays a triangle in all its glory, complete with labels I’ve provided for the sake of explanation (note in particular the angle between the two shorter sides, è). To find the trigonometric values of the triangle in Figure 2-1, you divide one side by another.
sin è = y/r
cos è = x/r
tan è = y/x
If you’re given the measure of one angle and one side of the triangle, you can
find all the others. Here are some examples — they’ll probably become distressingly familiar before you finish any physics course, but you don’t need to memorize them. If you know the preceding sine, cosine, and tangent equations, you can derive the following ones as needed:
x = r cos è = y/tan è
y = r sin è = x tan è
r = y/sin è = x/cos è
Remember that you can go backward with the inverse sine, cosine, and tangent, which are written as sin-1, cos-1, and tan-1. Basically, if you input the sine of an angle into the sin-1 equation, you end up with the measure of the angle itself.
sin-1(y/r) = è
cos-1(x/r) = è
tan-1(y/x) = è
h
x
y
è
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