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

Sig Figs The purpose of sig figs is to eliminate assumptions from your data. When measuring an object, we can only be as precise as our instrument enables us. For example, when using a ruler we can only read to the nearest mm or tenth of a cm, then estimate one more digit (tenth of a mm or hundreth of a cm).

Precision and Accuracy Accurate is getting a measurement similar to what the value should actually be Example: Shooting a basketball and hitting close to the rim every time (Desired result) Precise: means "repeatable, reliable, getting the same measurement each time." Example: Shooting a basketball and hitting the same spot on the backboard Precise and Accurate: Getting close to the same answer every time and getting close to the actual or accepted value Example: Shooting a basketball and making the shot each time without hitting the rim

Precise Precise and Accurate Completely random

Measuring with Accuracy and Precision In order to use an instrument accurately and precisely always estimate 1 decimal place pass the smallest increment

What is the diameter of the steel ball using the metric ruler and using correct sig figs?

2.15 cm

What is correct measurement of this triple beam balance?

What is correct measurement of this triple beam balance? 276.23 g

What is the correct measurement using the graduated cylinder and correct sig figs? Measure at the bottom of the meniscus.

21.6 ml

Rules for Sig Figs All non-zero numbers (1-9) are significant. Any zero between 2 significant digits are significant (sandwich rule) Any zero at the end of a decimal number is significant. Zeros that are used only as place holders are NOT significant (Zeros at the end of a whole number or beginning of a decimal number). If you need to make a zero significant, place a line over or under that zero. In scientific notation, all digits are significant.

Examples for Sig Figs all non-zero digits are 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 at the end of a decimal 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 or under that zero.  2 s.f.  3 s.f  3 s.f.

Multiplication & Division with Sig Figs Multiply and/or divide as indicated in the problem. Then round the answer to the same number of sig figs as the smallest number of sig figs of the numbers being multiplied or divided.

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.

Addition & Subtraction with Sig Figs Carry out the addition and/or subtraction as indicated in the problem. Then 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 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”

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.

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.

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.

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 of the independent variable. Choose a scale that fits your data. The x and y axis may have different scales. Make the graph as large as possible but use a convenient spacing that makes marking and reading the points easy. For example don’t mark 3 spaces between 0 and 10 since 3 1/3 is not a good scale. Remember that not all graphs will go through the origin (0,0).

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 independent variable on the horizontal (x) axis and the dependent variable on the vertical (y) axis. Plot every point. Title your graph. The title should state the purpose of the graph. It usually states the dependent vs the independent variable. Ex: velocity vs time

Best fit line or curve When there is a relationship between variables, draw a line (or curve) of best fit. It should be as close as possible to all of the data points. A good best fit line (or curve) should have the same number of points above and below the line (or curve) and be about the same distance from the line (or curve). If it is a straight line, use a ruler to draw it. Otherwise, draw a smooth curve.

Interpreting Graphs Most graphs that you will encounter in physics will be Linear Quadratic Inverse Inverse Square