Measurements, cont.

Significant figures:

For measurements we want to be sure we convey the precision (repeatability) of our measurements using significant figures. [covered in lab & Discussion problem set] You should note a couple of aspects of significant figures:

1. They are only used for measured quantities, and
2. You will be graded on them all semester, so learn them early and well, or you could lose a significant number of points!
• Look at example of making measurements with a ruler.
• Which digits should we keep? Which did we measure?

What is the reading (clicker question for last digit)? (0.18±0.01)

What is the measured value (clicker question for last digit)? (1.72±0.01)

Significant Figures and Calculations: Two basic sets of rules:

1. Addition/Subtraction rule: Significant figures are determined by looking at the decimal place of the numbers being added or subtracted. The number with the "least decimal places" determines the decimal place of the answer, e.g. if we add 1,216,956 to 214.879, the first number has the fewest decimal places, so the answer is rounded off to the 1's place: 1217170.879 goes to 1217171.
• Note that this is based on the idea that the error in the "least decimal place" measured figure is larger than the subsequent decimal figures, so they are dropped after rounding.
• examples.
• Note that with addition and subtraction we can end up with more or fewer significant digits in an answer.

2. Multiplication/Division rule: In this case we count the digits. The number with the fewest significant digits determines the number of significant digits in the answer.
• Note that in this case we are looking at how the error propagates as a fraction of the total (% error), for example
• for the problem 2.0 x 201 = 402 should be written as 2.0 x 201 = 4.0 x 102.
• This can be understood by doing the problem with upper and lower ranges of the measured number: 1.95<2.0<2.05
• 1.95 x 201 = 391.95
• 2.05 x 201 = 412.05
• Obviously we can't claim agreement with more than 2 significant figures. In fact, 4.0 is stretching it!
• examples.

Dimensional (Unit) Analysis and Problem Solving

A convenient check on your work, or even a way to determine the best approach to a problem, is to use dimensional analysis. This simply means to include all of the units for each factor in an equation, and then to check to see that the units on both sides of the equation are equal.

Example: How many centimeters are there in one foot?

Known: 1 ft = 12 inches (defined, therefore exactly); 2.54 cm = 1 inch (defined).

Set up: (1 ft)(12 inches/ft)(2.54 cm/inch)

note that ft cancels ft and inches cancels inches to give cm!

Solve: (1 ft)(12 inches/ft)(2.54 cm/inch) = 30.48 cm.

How about sig figs? In this problem there are no significant figures the way its set up, because there are no measurements! That is, all of the numbers are part of definitions, so they are exact, and that means the answer is exact as well.

 Extra Example: Assume it is 293 miles to your destination in the Bay Area. If you are driving a car that gets 27 miles per gallon and gas costs an average of \$3.45/gallon for the trip, how much will it cost you to drive there and back? Again, look at units. Want answer in dollars, so \$3.45/gallon ?? = \$ So we have \$ on both sides, but we need to get rid of (/gallon), so let's try dividing by 27 miles/gallon: (\$3.45/gallon) / (27 miles/gallon) ? = \$ We can now cancel gallons, as shown, and all we need to do now is multiply by miles to cancel miles: (293 miles)(\$3.45) / (27 miles) = \$ We now have the same units on each side, and can do the math, which I will leave to you.

NEXT

 Syllabus / Schedule C109 Home

© R A Paselk

Last modified 26 January 2015