CALCULATIONS & MEASUREMENT

Significant Digits

All digits that are certain plus one uncertain digit.

" 97.35 " has 4 significant digits

Facts about significant digits:

decimal points are not a factor in counting significant digits leading zeros are not counted as significant digits " 0.021 " has 2

significant digits

zeros that are bracketed by other digits are significant " 10.01" has 4 significant digits

zeros that follow other digits without decimals are not significant " 1200 " has 2 significant digits

Certainty Rule for Multiplying or Dividing

When multiplying or dividing, the answer has the same number of significant digits as the measurement with the fewest number of significant digits.

Example : 3.2 x 10.1 = 32.32 - the answer should only have 2 significant digits because of the certainty rule so it should be rounded to 32 which also has 2 significant digits.

Precision Rule for Adding or Subtracting

When adding or subtracting measured values of known precision, the answer has the same number of decimal places as the measured value with the fewest decimal places.

Example : 104.02 + 12 + 0.67 = 116.69 the answer should have no places after the units place so it would be represented as 117 .

Solving Equations

Defining equations are those equations that are used as quantity symbol representations of specific word definitions.

Example: D = M/V is the defining equation for density v = d/t is the defining equation for speed or velocity

When manipulating (rearranging) an equation you must keep the equation equal by performing the same functions on both sides of the equation.

Example :

v = d/t to solve for t v x t = d x t - on the right side the "t's " cancel leaving              t v x t = d - to isolate " t " divide both sides by v v x t = d - the " v's " on the left will cancel leaving   v      v t = d      v

The ultimate goal, when manipulating equations, is to have the variable that you are solving for, alone on the left side while all other parts of the equation remain on the right.

SOLVING PROBLEMS

Special attention is made here as to how to complete problems in steps.

1st - write down given quantities

d = ?

v = 5 m/s

t = 2 s

2nd - if any changes to the units need to be made, make them here.

3rd - write down the defining formula

v = d/t

4th - if the formula needs to be manipulated do so now

d = vt

5th - substitute the given values into the formula

d = 5 m/s x 2 s

6th - complete the calculation

d = 10 m

7th - make a statement including the correct units.

Problems like this are usually worth 6 or 7 points on a test or exam

Converting Units

cancel equivalent units top to bottom (Factor-label method) or simply by multiplying the number by a conversion factor that

could be memorized

t =  __ min x   1h                  60 min = 0.0166... h (this is the conversion factor for changing from minutes to hours)

1 m  x  60 s   x  60 min    x   1 k s     1min       1 h        1000 m

=  60 x 60 x 1 km ÷ 1000 h

=  3600 km ÷ 1000 h

= 3.6 km/h= 3.6 ( this is the conversion factor for changing from m/s to km/h)

For example  12 m/s =  12 x 3.6

                            = 43.2 km/h

You could divide by 3.6 if you wanted to convert km/h into m/s

For example    100 km/h =  100 ÷ 3.6

=27.77  m/s

Example 1 - If you were asked to change from 90 min to hours, complete the following:

90 min x 0.0166... = 1.5 h

Example 2 - If you were asked to change from 60 km/h to m/s

60 x 0.2777 or 60 / 3.6 = 16.6 m/s

Example 3 - If you were asked to change from 40 m/s to km/h

38.9 x 3.6 = 140.04 km/h …. 140 km/h