Units and Measurements 1.2 PDF

UNITS AND MEASUREMENTS LESSON 1.2 GENERAL PHYSICS I  Accuracy and Precision Accuracy is how close you are to the actual value and depends on the person measuring or refers to how close measurements are to the "true" value Example: We want to measure the length of...

UNITS AND MEASUREMENTS LESSON 1.2 GENERAL PHYSICS I  Accuracy and Precision Accuracy is how close you are to the actual value and depends on the person measuring or refers to how close measurements are to the "true" value Example: We want to measure the length of standard printer paper. The packaging in which we purchased the paper states that it is 11.0. in. long. We then measure the length of the paper three times and obtain the following measurements, 11.1 in., 11.2 in., and 10.9.in. These measurements are quite accurate because they are very close to the reference value of 11.0 in. Notice that the concept of accuracy requires that an accepted reference value be given.  Accuracy and Precision Precision refers to how close measurements are to each other or a measurement system refers to how close the agreement is between repeated measurements (which are repeated under the same conditions). Consider the example of the paper measurements. The precision of the measurements refers to the spread of the measured values. One way to analyze the precision of the measurements would be to determine the range, or difference, between the lowest and the highest measured values. In that case, the lowest value was 10.9 in. and the highest value was 11.2 in. Thus, the measured values deviated from each other by at most 0.3 in. These measurements were relatively precise because they did not vary too much in value. However, if the measured values had been 10.9, 11.1, and 11.9, then the measurements would not be very precise because there would be significant variation from one measurement to another.  Difference Between Accuracy and Precision ACCURACY PRECISION Accuracy refers to the level of Precision implies the level of agreement between the actual variation that lies in the values of measurement and the absolute several measurements of the same measurement. factor. Represents how closely the results Represents how closely results agree with the standard value agree with one another Single-factor or measurement Multiple measurements or factors are needed It is possible for a measurement to Results can be precise without being be accurate on occasion as a fluke. accurate. Alternatively, the results For a measurement to be can be precise and accurate. consistently accurate, it should also Accuracy and Precision  Significant Figures Significant figures are used to determine the accuracy and precision of measuring devices. A device that can report its measurements with more significant figures is said to be more accurate. When scientists combine their measurements in a mathematical equation to derive some other values, the final result usually ends up with many decimal places. For example, when converting 100 pounds to kilograms, the result is 45.45kg, with the last two digits repeating. The last digit 5 is understood to be uncertain or estimated. When converting 98֯ F to the Celsius scale, we end up with the value of 36.6666666֯C. The repeating digit 6 after the decimal point can be rounded off. This gives us a temperature of 36 degrees to be certain and 0.6 degrees Celsius as estimated.  Rules for Significant Figures 1. Any digit that is not zero is significant 543 m – 3 significant figures 5.43 L– 3 significant figures 2. Leading a zero is never significant. Leading zeros are located to the left of the first nonzero digit. They simply show where the decimal point is located. 0.00004 mm – 1 significant figures 0.4 cm – 1 significant figures 3. Zeros in between significant digits are always significant are always significant. 3 002 mg – 4 significant figures 60.0508 kg – 6 significant figures 4. Trailing zero is significant only if the decimal point is identified or specified a. When there is no decimal point, the last zeros are not significant as they only indicate where the decimal point is to be located. 2 020 days – 3 significant figures 70 cm3 – 1 significant figures b. When the decimal point is indicated. 1. For numbers greater than 1, all the trailing zeros to the right of the decimal point are significant. 2 020.0 s – 5 significant figures 707.0000 min – 7 significant figures 2. For numbers less than 1, only the trailing zeros at the end of the number or the zeros in between nonzero digits are significant. 0.07080 km – 4 significant figures 0.010 mg – 2 significant figures  Rules in Rounding Off Numbers 1. When the number to be dropped off is greater than 5, round off by adding 1 to the last digit to be retained. Rounded off to two significant figures 235 g = 240 g Rounded off to one significant figure 785 cm = 800 cm 2. When the number to be dropped off is less than 5, simply drop off the digit and replace with a zero; do not add any number to the last digit to be retained. Rounded off to three significant figures 4 123 kg = 4 120 kg Rounded off to two significant figures 234 s = 230 s 3. There are two rules when the digit to be dropped off is equal to 5. a. If the digit to be retained is even, then just drop 5 if it appears after the decimal point or replace with zero if it appears before the decimal point. (The even number to be retained is underscored) Round off to four significant figures 362.45 N = 362.5 N Round off to one significant figures 705 Ohms = 700 ohms Note: Zero is an even number b. When the digit to be retained is odd, add 1 to the digit to be retained and drop the 5 and all the numbers that follow if it appears after the decimal point or replace with zero if it appears before the decimal point. (The odd number to be retained is underscored) Rounded off to three significant figures 0.37754 A = 0.378 A Rounded off to two significant figures 875 J/s = 880 J/s  Scientific Notation Science is also works with very large and very small numbers. To make it easier to work with these numbers, write them in scientific notation. Numbers written in scientific notation is composed of two parts. The first part called digit term is a number from 1 to 9 and the second part, a power of ten, is called exponential term. An atomic mass unit (amu) is approximately 000 000 000 000 000 000 000 001 660 54 kg Which in scientific notation is 1.660 54 x 10 -24. The digits 166 054 are the significant digits of the number. The decimal point is placed immediately after the first significant digit. Note the direction and the count the number of times the decimal point is moved from its origin position. Since it moves from left to right, the exponent is negative.  From Standard Notation to Scientific Notation To rewrite a number using scientific notation: 1. Copy the significant digits of the number 2. Identify the first significant digit and move the decimal point to its right. 3. Count the number of places the decimal point is moved. This will be the power or exponent. 4. Note the direction of the decimal point is moved. This will be determine the sign of the exponent. If the decimal point is moved to the left, then the exponent is positive. If the decimal point is moved to the right, then the exponent is negative. Example: 6. 980 000 000 = 6.98 x 109 1. The significant digits are 6, 9, and 9. Place the decimal point after the 1st significant digit (6). The decimal point has moved nine places to the left. Therefore, the sign of the exponential term is positive (109) 0.00698 = 6.98 x 10-3 2. The significant digits are 6. 9 and 8. Place the decimal point after the 1 st significant digit (6). The decimal point has moved three places to the right. Therefore, the sign of the exponential term is positive (10-3)  From Scientific Notation to Standard Notation 1. The sign of the exponent indicates the directions in which to move the decimal point. When rewriting the numbers from scientific notation to standard notation, the rules are simply reversed. A positive exponent means that the number is less than 1, hence, we move the decimal point to the left. 2. The exponent tells us how many places we have to move the decimal point. Example: 9.30 x 106 Move the decimal point six places to the right 9 300 000 Standard Notation 8 x 10 -2 If the decimal point is not indicated, it is assumed to be at the end of the number. Move the decimal places to the left. 0.08 Standard Notation