Physics Basics: Understanding Sig Figs, Dimensional Analysis, and Graphing Rules

1. Physics BasicsFirst Week of School Stuff Sig. Figs, Dimensional Analysis, Graphing Rules, Techniques, and Interpretation

2. Sig Figs • The purpose of sig figs is to eliminate assumptions from your data. When measuring an object, we can only be as accurate as our instrument enables us. For example, when using a ruler we can only read to the nearest mm, then estimate one more digit. When doing calculations sig figs are determined by our most inaccurate number.

3. Rules for Sig Figs • All non-zero numbers are significant • Any zero between 2 significant digits are significant (sandwich rule) • Any zero after the decimal at the end of a number are significant • Zeros that are used only as place holders are NOT significant • If you need to make a zero significant when it is normally not, place a line over that zero.

4. Examples for Sig Figs all digits are non-zero and therefore significant 1, 245 4 sig figs 347.58  5 sig figs 23  2 sig figs zeros between two sig figs are significant (sandwich rule) 102 3 sig figs 13.505 5 sig figs 23060.7  6 sig figs zeros after the decimal at the end of a number are significant 1, 245.0 5 sig figs 2.30  3 sig figs 347.580  6 sig figs zeros that are used only as place holders are NOT significant 120 2 sig figs 0. 0153  3 sig figs 0.00304  3 sig figs 0.00250400  6 sig figs to make a zero significant when it is normally not, place a line over that zero.  2 s.f.  3 s.f  3 s.f.

5. Multiplication & Division with Sig Figs • First, count the number of sig figs in each number to be multiplied or divided and make note of the smallest number of sig figs. • Then carry out the multiplication or division as indicated in the problem • Finally, round the answer to the number of sig figs that is the same as the number multiplied or divided with the smallest number of sig figs to begin with.

6. Example: Mult. or Div. 3459 x 25 86,475 86,000 4 sig figs 3654 ÷ 151 = 24.1986755  2 sig figs 4 s.f. 2 s.f. Ans. w/o rounding  Answer without rounding  Answer rounded to the proper number of sig figs 3654/151 = 24.2 According to Sig Fig rules the answer must have the same number of sig figs as the number with the least sig figs…in this case  2 sig figs, so now you need to round the answer to 2 sig figs Answer must be rounded to 3 sig figs because 151 has the least s.f. with 3.

7. Addition & Subtraction with Sig Figs • First, decide what the precision of each number is (What place value is each number expressed to?) • Ex: 2351 is precise to the ones place and 14.7 is precise to the tenths • Then carry out the addition or subtraction as indicated in the problem • Finally, round the answer to the place value of the least precise number you began with. • In our example above: the answer to an addition or subtraction with these numbers would be rounded to the “ones” place because “ones” is a less precise measurement than “tenths”

8. Example: Add & Subtract 345.3876 - 6.45__ 338.9376  “ten-thousandths” 258.2 + 6.539 264.739 to the “tenths”  “hundredths” to the “thousandths”  answer without rounding  answer without rounding According to sig fig rules, answer should be rounded to the “tenths” place because that is less precise than “thousandths” According to sig fig rules, answer should be rounded to the “hundredths” place because that is less precise than “ten-thousandths” So…answer is 264.7 for proper sig figs. So…answer is 338.94 for proper sig figs.

9. Dimensional Analysis • Dimensional Analysis is a simple way of converting from one unit to another. • It is also a way of solving problems using units as a guide to the mathematical steps necessary.

10. Dimensional Analysis • Example: To go from 100km per hour to cm per sec we would begin by writing out 100km over 1 hour. Then we would multiply it by the number of meters in 1 km and then by the number of cm in 1 meter. Now we need to convert to seconds. We would divide the whole answer by 60 to get to minutes and then again to get it to seconds.

11. Graphing Rules and Techniques • Identify the dependent and independent variables. The independent variable is the one that is intentionally changed while the dependent variable changes as a result to the alteration of the independent variable. • Choose the scale you will use for the graph. Make the graph as large as possible and use a convenient spacing. For example, dividing ten into four spaces is not as simple as dividing it into five. • Remember that not all graphs will go through the origin (0,0).

12. More Graphing Rules • Each axis should be labeled with the name of the variable and the units you are using. Make the axis darker as to be easily identifiable. Plot the dependent variable on the horizontal (x) axis and the independent variable on the vertical (y) axis. Plot every point. • Title your graph. The title should state the purpose of the graph and include all the dependent and independent variables.

13. In some instances, such as acceleration and velocity, you may get data points that are, or appear to be in a straight line. When that happens, draw a straight line going through as many points as possible and having the same number of points above and below the line.

14. Interpreting Graphs • Most graphs that you will encounter in physics either A) have the dependent variable vary directly to the independent variable B) Vary inversely to the other variable or C) vary directly to the square of the other variable.