Physics I For Dummies. Steven Holzner

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СКАЧАТЬ in 7.0 seconds, the person is telling you that the distance is known to three significant digits and the seconds are known to two significant digits (the number of digits in each of the measurements). If you want to find the rocket’s speed, you can whip out a calculator and divide 10.0 meters by 7.0 seconds 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 two or 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. You need to round your answer.

The rules for determining the correct number of significant digits after doing calculations 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 (the number of significant digits in 7.0). The best you can say is that the rocket is traveling at 1.4 meters per second, which is 1.428571429 rounded to one decimal place.

       When you add or subtract numbers: Line up the decimal points; the last significant digit in the result corresponds to the right-hand 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. You can see what we mean by taking a look for yourself:

       When you round the answer to the correct number of significant digits, your answer is 24.

      

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, round up. If it’s 4 or less, round down. For example, you round 1.428 up to 1.43 and 1.42 down to 1.4.

      Estimating accuracy

      Physicists don’t always rely on significant digits when recording measurements. Sometimes, you see measurements that use plus-or-minus signs to indicate possible error in measurement, as in the following:

      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

(that is, 5.41) meters and

(that is, 5.31 meters), inclusive. Note that the possible error isn’t the amount your measurement differs from the “right” answer; it’s an indication of how precisely your apparatus can measure — in other words, how reliable your results are as a measurement.

Arming Yourself with Basic Algebra

      Physics deals with plenty of equations, and to be able to handle them, you should know how to move the variables in them around. Note that algebra doesn’t just allow you to plug in numbers and find values of different variables; it also lets you rearrange equations so you can make substitutions in other equations, and these new equations show different physics concepts. If you can follow along with the derivation of a formula in a physics book, you can get a better understanding of why the world works the way it does. That’s pretty important stuff! Time to travel back to basic algebra for a quick refresher.

      You need to be able to isolate different variables. For instance, the following equation tells you the distance, s, that an object travels if it starts from rest and accelerates at rate of a for a time, t:

      Now suppose the problem actually tells you the time the object is in motion and the distance it travels and asks you to calculate the object’s acceleration. By rearranging the equation algebraically, you can solve for the acceleration:

      In this case, you’ve multiplied both sides by 2 and divided both sides by t² 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:

      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. (If you need a review of algebra, get a copy of Algebra I For Dummies by Mary Jane Sterling [Wiley].)

Tackling a Little Trig

      You need to know a little trigonometry, including the sine, cosine, and tangent functions, for physics problems. To find these values, start with a simple right triangle. Take a look at Figure 2-1, which displays a right triangle in all its glory, complete with labels we’ve provided for the sake of explanation. Note in particular the angle

, which appears between one of the triangle’s legs and the hypotenuse (the longest side, which is opposite the right angle). The side y is opposite

, and the side x is adjacent to

.

      FIGURE 2-1: A labeled triangle that you can use to find trig values.

      

To find the trigonometric values of the triangle in Figure 2-1, you divide one side by another. Here are the definitions of sine, cosine, and tangent:

       

       

       

      If you’re given the measure of one angle and one side of the triangle, you can find all the other sides. Here are some other forms of the trig relationships — 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:

       

       

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