Lecture notes #1, Physical Measurements  201 Lab:  Dr. Erkal

Significant figures:

In any experiment, the measured quantities are only known to within the limits of the experimental uncertainty.  The value of the uncertainty can depend on such factor as

the quality of the apparatus, the skill of the experimenter, and the number of measurements performed.  Assume we are measuring the area of a rectangular plate using meter stick that has an accuracy of ±1cm.  If the length of the plate is measured as 16.3 cm, we can only claim that that its length lies somewhere between 16.2 cm and 16.4 cm.  Suppose the width is 4.5 cm.  We could express the results of the measurements as

           

            a=16.2 ± 0.1

            b = 4.5 ± 0.1

 

Area= axb=16.3cmx4.5cm=73.35 cm2.

If we claim that the area is 73.35 cm2, it would be unjustifiable since it contains four SF.

Here are the rules to keep track of SF during calculations:

 

1)      When multiplying or dividing several quantities, the number of SF in the final answer is the same as the number of SF in the least accurate of the quantities being multiplied (or divided).  Least accurate means, “having the lowest number of SF”.

2)      For addition and subtraction, the number of palaces must be considered instead of SF.  When numbers are added or subtracted, the number of places in the result should equal the smallest number of decimal places of any term in the sum.

3)      Multiplying or dividing a number by any exact constant usually doesn’t change the number of SF.

4)      How many SF should we include when we quote an error?  Usually, one SF is acceptable.  WE will talk about the rules that come into the play in error analysis.

 

 

Relative Error in A Quantity:

 

 

Example:

Error in measuring with meter stick.

 

Smallest division is 1 mm or 0.1 cm.  Meter stick is 100 cm. Therefore,

 

Relative error = 0.1/100=0.001, Percentage error = 100 x relative error = 0.1%

 

 

Rounding off:

 

When we are finished with a calculation as result of processing a number of values in a formula, we often like to give the result to a certain number of SF.  Once the correct number of SF is determined, the next digit in the value is examined.  If the next digit is greater than “5”, the value is reported, then the preceding digit is increased by one.  If the next digit is is less than “5”, the value is reported to the number of SF with no change.  If the next digit is ‘5’, the last significant figure is made an even number-rounding up if necessary.

 

Scientific Notation:

Sometime the presence of zeros may be misinterpreted.  For example, suppose the mass of an object is measured to be 1500g.  This value is ambiguous in that it is not known whether the last two zeros are being used to locate the decimal point or they represent significant figures in the measurement.  We can remove this ambiguity by using scientific notation which uses powers of ten.  In this case 1500 g should be expressed as

 

1.50 x103 g if three significant figures are measured.

1.5 x 10 3 g if two significant figures are measured.

 In scientific notation, only significant digits are written with one digit ( non-zero) before the decimal and with the appropriate powers of ten as a multiplier.

 

Examples:

 

0.4728        is 4.7 x 10-2 in 2 SF, and 4.73 x 10-2 in 3 SF

 

392.6   is 3.9 x 10 2 in 2 SF    and 3.93 x 10 2 in 3 SF