LECTURER PROF.Dr. DEMIR BAYKA AUTOMOTIVE ENGINEERING LABORATORY I

UNCERTAINTY ANALYSIS ERROR IT IS THE DIFFERENCE BETWEEN THE MEASURED AND TRUE VALUE THE TRUE VALUE MUST BE KNOWN IN ORDER TO CALCULATE THE ERROR

UNCERTAINTY ANALYSIS SINCE THE TRUE VALUE IS UNKNOWN UNCERTAINTY UNCERTAINTY IS THE ESTIMATED ERROR IT IS WHAT WE THINK THE ERROR IS

PRECISION ERROR IT IS PRESENT WHEN SUCCESSIVE MEASUREMENTS OF AN UNCHANGED QUANTITY YIELDS NUMERICALLY DIFFERENT VALUES UNCERTAINTY ANALYSIS

ACCURACY ERROR IT IS PRESENT WHEN THE NUMERICAL AVERAGE OF SUCCESSIVE READINGS DEVIATES FROM THE KNOWN CORRECT READING UNCERTAINTY ANALYSIS

ACCURACY ERRORS MAY BE CORRECTED (FOR EXAMPLE BY CALIBRATION) PRECISION ERRORS HAVE TO BE CALCULATED IN A STATISTICAL MANNER UNCERTAINTY ANALYSIS

WHEN A MEASURED QUANTITY ( X ) IS UNCERTAIN THEN THE RESULT IS PRESENTED IN THE FOLLOWING MANNER : UNCERTAINTY ANALYSIS

IF A MEASUREMENT OF A PHYSICAL QUANTITY IS EXPRESSED WITHOUT ANY UNCERTAINTY THE UNCERTAINTY IS DEDUCED FROM THE SIGNIFICANT FIGURES UNCERTAINTY ANALYSIS

SIGNIFICANT FIGURES THE NUMBER OF FIGURES USED IN EXPRESSING THE RESULTS OF A MEASUREMENT IS AN INDICATION OF THE ACCURACY OF THAT MEASUREMENT UNCERTAINTY ANALYSIS

EXAMPLE IF THE MASS OF AN OBJECT IS SPECIFIED AS 12 Kg m = 12 kg THEN THE TRUE VALUE IS ESTIMATED TO BE CLOSER TO 12 kg THAN TO 11 kg OR 13 kg

IF THE SAME MASS WAS MEASURED BY A MORE ACCURATE SCALE AS : m = 12.0 kg THEN ITS TRUE VALUE IS ESTIMATED TO BE CLOSER TO 12.0 kg THAN TO 12.1 kg OR 11.9 kg

THERE ARE 2 SIGNIFICANT FIGURES m = 12 kg IN m = 12.0 kg IN THERE ARE 3 SIGNIFICANT FIGURES

FOR WHOLE NUMBERS THE SIGNIFICANT FIGURES ARE CONSIDERED TO BE INFINITE FOR EXAMPLE IN THE FORMULA THE 2 IN THE DENOMINATOR IS CONSIDERED TO BE EXACT (==> …..)

FOR 1.TRAILING ZEROES E.g. 600 OR OR LEADING ZEROES E.g OR OR 0.12 TOTAL NUMBER OF DIGITS MAY NOT ALWAYS CORRESPOND TO NUMBER OF SIGNIFICANT FIGURES

IN ORDER TO CLEARLY EXPRESS THE NUMBER OF SIGNIFICANT FIGURES USE SCIENTIFIC NOTATION FOR EXAMPLE INSTEAD OF X = 2300 X = 2.3 x 10 3 ( 2 SIGNIFICANT FIGURES) X = 2.30 x 10 3 ( 3 SIGNIFICANT FIGURES) X = x 10 3 ( 4 SIGNIFICANT FIGURES)

IF ONLY SIGNIFICANT FIGURES ARE USED TO EXPRESS THE MEASURE OF A QUANTITY THE UNCERTAINTY IS IN THE LEAST SIGNIFICANT DIGIT

IF THE UNCERTAINTY IN THE MEASURE OF A QUANTITY IS OTHER THAN THE LEAST SIGNIFICANT DIGIT THEN IT IS EXPLICITLY EXPRESSED AS : *ABSOLUTEUNCERTAINTY *RELATIVEUNCERTAINTY

COMPUTATIONS OF NUMBERS HAVING UNEQUAL NUMBER OF SIGNIFICANT FIGURES ADDITION AND SUBTRACTION RESULT IS EXPRESSED WITH AN ACCURACY EQUAL TO THE ACCURACY OF THE LEAST ACCURATE NUMBER.

COMPUTATIONS OF NUMBERS HAVING UNEQUAL NUMBER OF SIGNIFICANT FIGURES ADDITION AND SUBTRACTION for example

COMPUTATIONS OF NUMBERS HAVING UNEQUAL NUMBER OF SIGNIFICANT FIGURES ADDITION AND SUBTRACTION for example

COMPUTATIONS OF NUMBERS HAVING UNEQUAL NUMBER OF SIGNIFICANT FIGURES ADDITION AND SUBTRACTION for example

COMPUTATIONS OF NUMBERS HAVING UNEQUAL NUMBER OF SIGNIFICANT FIGURES MULTIPLICATION AND DIVISION RESULT IS EXPRESSED WITH AN ACCURACY EQUAL TO OR LESS THAN THE ACCURACY OF THE LEAST ACCURATE NUMBER.

for example COMPUTATIONS OF NUMBERS HAVING UNEQUAL NUMBER OF SIGNIFICANT FIGURES MULTIPLICATION AND DIVISION

for example COMPUTATIONS OF NUMBERS HAVING UNEQUAL NUMBER OF SIGNIFICANT FIGURES MULTIPLICATION AND DIVISION

COMBINATION OF UNCERTAINTIES AS WELL AS ADDITION, SUBTRACTION, MULTIPLICATION AND DIVISION THERE WILL ALSO BE CASES WHERE POWERS OF VARIABLES WILL BE COMBINED TO OBTAIN A RESULT.

COMBINATION OF UNCERTAINTIES THE UNCERTAINTY OF THE RESULT WILL DEPEND : *ON THE UNCERTAINTITIES OF THE INDIVIDUAL MEASURED QUANTITIES *ON HOW THESE QUANTITIES ARE COMBINED

COMBINATION OF UNCERTAINTIES IN GENERAL IF RESULT Q IS A FUNCTION OF MORE THAN ONE VARIABLE x i THEN THE EXPECTED VALUE Q WILL BE CALCULATED THROUGH THE EXPECTED VALUES OF THE AFFECTING VARIABLES x i AND WILL HAVE AN OVERALL UNCERTAINTY

COMBINATION OF UNCERTAINTIES WILL BE DEPENDENT ON AND ON HOW MUCH EACH AFFECTS i.e.

BY ADDING THE SQUARES OF RELATIVE UNCERTAINTY EFFECT OF EACH VARIABLE AND THEN TAKING THE SQUARE ROOT OF THE SUM WE ARE CONSIDERING THE EFFECT OF EACH VARIABLE IRRESPECTIVE OF ITS TREND

THE VARIABLES ( x i ) ARE ASSUMED TO BE INDEPENDENT

WE MAY CONSIDER AS WEIGHING FACTORS THAT DETERMINE THE RELATIVE EFFECTS OF ON

EXAMPLE V1V1 V2V2 R

V1V1 V2V2 R EXPECTED VALUE IT IS POSSIBLE THAT

V1V1 V2V2 R WOULD GIVE US

V1V1 V2V2 R HOWEVER WE ARE LESS CERTAIN OF BOTH * THE MEASURED VOLTAGE AND * THE MEASURED OR ACCEPTED RESISTANCE

THEREFORE FOR

THEREFORE

EXAMPLE

Carbon dioxide is pumped into a cylinder. The pressure and temperature of the gas is measured after they have reached stable values. Find the mass of the gas with its uncertainty. Measured data : D = 20 cm±0.1 cm H = 100 cm±0.5 cm P = 100 atm±1 atm T = 20 C±0.1 C

Given D = 20 cm± 0.1 cm H = 100 cm± 0.5 cm P = 100 atm± 1 atm T = 20 C± 0.1 C Required m = ?  m = ? Solution