 # Physical quantities and their measurements

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Physical quantities and their measurements
Measurement of length and volume Instruments of measurement Least count and Precision of an instrument Significant figures Uncertainty in a measurement Accuracy and errors – Systematic and random

Measuring Volume of a liquid http://www. chem. tamu

Beaker – least count and significant figures
The smallest division is 10 mL, so we can read the volume to 1/10 of 10 mL or 1 mL. The volume we read from the beaker has a reading error of 1 mL. The volume in this beaker is mL. You might have read 46 mL; your friend might read the volume as 48 mL. All the answers are correct within the reading error of 1 mL. So, How many significant figures does our volume of mL have? Answer - 2! The "4" we know for sure plus the "7" we had to estimate.

Graduated cylinder First, note that the surface of the liquid is curved. This is called the meniscus. This phenomenon is caused by the fact that water molecules are more attracted to glass than to each other (adhesive forces are stronger than cohesive forces). When we read the volume, we read it at the BOTTOM of the meniscus. The smallest division of this graduated cylinder is 1 mL. Therefore, our reading error will be 0.1 mL or 1/10 of the smallest division. An appropriate reading of the volume is mL. An equally precise value would be 36.6 mL or 36.4 mL. How many significant figures does our answer have? 3! The "3" and the "6" we know for sure and the "5" we had to estimate a little.

Buret The smallest division in this buret is 0.1 mL. Therefore, our reading error is mL. A good volume reading is mL. An equally precise answer would be mL or mL. How many significant figures does our answer have? 4! The "2", "0", and "3" we definitely know and the "8" we had to estimate.

Conclusion – sf and precision
The number of significant figures is directly linked to a measurement. If a person needed only a rough estimate of volume, the beaker volume is satisfactory (2 significant figures), otherwise one should use the graduated cylinder (3 significant figures) or better yet, the buret (4 significant figures). So, does the concept of significant figures deal with precision or accuracy? Hopefully, you can see that it really deals with precision only.

Rulers Meter rule in inch, centimetre and millimetre Half metre rule
Precision is how fine of a measurement that the measuring instrument is marked off for.

Rulers in inches

Standard English Ruler

Standard ruler Metric

Least Count The least count is the smallest subdivision marked on a measuring instrument. Determine the least count of your measuring stick. Record the numerical value of the least count and the unit of measurement. Example: A meter stick is divided into 100 equal divisions and numbered. Each of these numbered divisions is called 1 cm. ('one centimeter' means 'one one-hundredth' of a meter). Each centimeter is further divided into 10 equal divisions. This is the smallest subdivision on the meter stick.

Rounding off and significant figures

Other instruments to measure length
Micrometer screw gauge Vernier calipers

Rules for Working with Significant Figures:
Leading zeros are never significant. Imbedded zeros are always significant. Trailing zeros are significant only if the decimal point is specified. Hint: Change the number to scientific notation. It is easier to see. Addition or Subtraction: The last digit retained is set by the first doubtful digit. Multiplication or Division: The answer contains no more significant figures than the least accurately known number.

Exact and inexact numbers
example: There are exactly 12 eggs in a dozen. example: Most people have exactly 10 fingers and 10 toes.

inexact numbers: example: any measurement. If I quickly measure the width of a piece of notebook paper, I might get 220 mm (2 significant figures). If I am more precise, I might get 216 mm (3 significant figures). An even more precise measurement would be mm (4 significant figures).

Significant figures 0.00682 3 6.82 x 10-3 1.072 4 1.072 (x 100)

significant figures

Significant digits 3 sig. digs. .5 sig. digs. 1 sig. dig. 2 sig. digs. 4 sig. digs.

Rules for Working with Significant Figures:1
Leading zeros are never significant. Imbedded zeros are always significant. Trailing zeros are significant only if the decimal point is specified. Hint: Change the number to scientific notation. It is easier to see.

Rules for Working with Significant Figures: 2
Addition or Subtraction: The last digit retained is set by the first doubtful digit. Multiplication or Division: The answer contains no more significant figures than the least accurately known number.

Rounding off Addition or Subtraction: The last digit retained is set by the first doubtful digit

Addition or Subtraction: The last digit retained is set by the first doubtful digit

Multiplication or Division:The answer contains no more significant figures than the least accurately know number.

Multiplication or Division: The answer contains no more significant figures than the least accurately known number.

Significant figures and electronic calculators
. For example, dividing 5.0 by 1.67 on a calculator may give the following answer: 5.0 / 1.67= The correct answer, 3.0, has only two significant figures, as in the least accurate number (5.0) in the problem. All other digits displayed by the calculator are insignificant.

Rounding off Let's round off 64,492 to: (a) 1 significant figure      which is 60,000 (b) 2 significant figures     which is 64,000 (c) 3 significant figures     which is 64,500 (d) 4 significant figures     which is 64,490 (e) 5 significant figures     which is 64,492

rounding off to given significant figures
When rounding off numbers to a certain number of significant figures, do so to the nearest value. example: Round to 3 significant figures: x 104 (Answer: 2.35 x 104) example: Round to 2 significant figures: x 103 (Answer: 1.6 x 103)

What happens if there is a 5? There is an arbitrary rule:
If the number before the 5 is odd, round up. If the number before the 5 is even, let it be. The justification for this is that in the course of a series of many calculations, any rounding errors will be averaged out. example: Round to 2 significant figures: 2.35 x 102 (Answer: 2.4 x 102) example: Round to 2 significant figures: 2.45 x 102 (Answer: 2.4 x 102) Of course, if we round to 2 significant figures: x 102, the answer is definitely 2.5 x 102 since x 102 is closer to 2.5 x 102 than 2.4 x 102.

QUIZ: Question 1 Give the correct number of significant figures for 4500, 4500., , Question 2 Give the answer to the correct number of significant figures: = ?

Questions Question 3 Give the answer to the correct number of significant figures: = ? Question 4 Give the answer to the correct number of significant figures: (1.3 x 103)(5.724 x 104) = ?

Questions Question 5 Give the answer to the correct number of significant figures: (6305)/(0.010) = ?

Example Significant figures
Number Exponential expression Significant figures 560, X two (The zeros show only the location of the decimal point.) 560, X 105 six (The decimal point in the original number shows that all the zeros are significant.) 30, X four (The first zero is between two digits and is significant. The last shows only the location of the decimal point.)

X three (The first two zeros show the location of the decimal; they are not significant. The last one does not show the location of the decimal point; it reports a measurement and therefore is significant.)

Errors Random Errors: Errors that can not be predicted.
Systematic Errors: Errors which are the same for each measurement

Precision and accuracy
Poor Precision -> the hits are not near each other Poor Accuracy -> the hits are not near their intended target Good Precision -> all the hits are close to each other Good Accuracy -> all the hits are near their intended target Good Precision -> all the hits are close to each other Poor Accuracy -> the hits are not near their intended target

Precision: Small random error
Accurate: Small systematic error

Absolute uncertainty Size of an error and its units
30m tape measure has an error of ±0.5cm So the Absolute error is 30±0.005 m

Fractional uncertainty
Absolute uncertainty / measurement 30m tape measure has an error of ±0.5cm Fractional Uncertainty = 0.005/30

Percentage uncertainty
Fractional uncertainty x 100% Therefore 0.005/30 x 100% = 0.017%

5.9 ±0.6m ±0.8m = ±1.4m (add absolute errors) 6.9 ±0.6m ±0.8m = 3.0 ±1.4m (add absolute errors)

Multiplication and Division
5.6 ±0.5m x 2.6 ±0.5m = 15 ±??m 0.5 / 5.6 = / 2.6 = Sum of relative errors = Absolute error = x 15 = 4.2m FINAL ANSWER = 15 ±4 m

Example Question The length of a piece of paper is measured as 297± 1mm. Its width is measured as 209± 1mm What is the fractional uncertainty in its length What is the percentage uncertainty in its length? What is the area of one side of the piece of paper? State your answer with its uncertainty

2.) What is the area of your desk?
Remember A/A = L/L + W/W

Example Question 2 A Voltmeter has a reading of 2.00±0.05 V, a miliammeter reads 3.3±0.1 mA Estimate the Resistance and state your uncertainties.

Question 3 What do the terms systematic and random errors mean?