1.2 Measurement and Uncertainties

1.2 Measurement and Uncertainties

1.2.1 Fundamental Units

All science uses SI units – Le Systeme International d’Unites(The International System of Units)

(ex) mass kg (kilograms)

force N (newtons) These are the SI units for mass, force, and energy

energy J (joules)

Physics is the most fundamental science. It measures physical properties of different things, compares them to reference quantities called fundamental quantities, and expresses the answer in numbers and units.
Fundamental Quantities/Units -» The basic quantities in physics, which can’t be measured in a simpler form. These units can be used to generate all other units. There are seven fundamental quantities:

Question: How do you define “one meter”?

Answer: One meter is defined as the length of path travelled by light in a vacuum during a time interval of 1/299 792 453 seconds.

A good standard measurement should be constant (invariant in time), and reproducible so that it can be used all over the world.
Most standard measurements are based on properties of atoms.

(ex) One second is the time for 9 192 631 770 vibrations of the cesium-133 atom.

When we measure things, we should make sure to use an appropriate instrument.

(ex) We would not use the same instrument to measure the size of an atom as we would to measure the height of a person.

1.2.2 Fundamental and Derived Units

When a quantity is found by measuring two or more fundamental quantities, it is called a derived quantity, and its units are called derived units.
Derived Quantities/Units -» Quantities or units that are made up of two or more fundamental quantities or units.

(ex) velocity m s-1 (metres per second)

acceleration m s-2 (metres per second-squared)

momentum kg m s-1 (kilogram metres per second)/ or N s (newton seconds)

The following table shows some of the most common derived units:

Question: Which one of the following lists a fundamental unit followed by a derived unit?

A. amperemole

B. coulombwatt

C. amperejoule

D. secondkilogram

Answer: C.

1.2.3 Conversion between Different Units

Sometimes we can express quantities using different derived units. A conversion is a change from one derived unit to another.

(ex) Force:kg m s-2 = N (newton)

Energy:J (joule) = W h (watt hour) = N m (newton metres) = eV (electron volt)

Power:W (watt) = kg m2 s-3 = J s-1

1.2.4 Units in Accepted SI Format

Proper SI units use, for example, m s-1, not m/s.

1.2.5 Scientific Notation and Prefixes

Really large or really small numbers can be very difficult to write. Scientific Notation is an easier, standard way to write numbers.

(ex 1) 1 200 000 = 1.2 x 106

106 = 1 000 000 , so

1.2 x 106 = 1.2 x 1 000 000 = 1 200 000 = 1.2 x 106

(ex 2) 0.06 kg = 6 x 10-2 kg

Question: What is the scientific notation for 132.97 kg and 1 401.2 kg?

Answer: 132.97 kg = 1.3297 x 102 kg, 1 401.2 kg = 1.4012 x 103 kg.

For certain numbers, it is easier to use prefixes as well as scientific notation.

(ex) 1 000 000 000 m = 1 x 109 m = 1 Gm (one gigametre)

This is a list of common prefixes:

Question: How would you write 123 000 J using scientific notation and prefixes?

Answer: 123 000 J = 123 kJ = 1.23 x 102 kJ

HOMEWORK: Physics 3rd edition pg 6-7, # 1 – 11

1.2.6 Random and Systematic Errors

No experiment is perfect. When we do experiments, there are many different errors that can happen. There are two types of errors: random errors and systematic errors.
Systematic Error -» An error which causes a random set of measurements to be spread about an incorrect value rather than being spread about the accepted value.

(ex) badly made instruments

Poorly calibrated instrumentsexamples of sources of

an instrument having a zero(calibration) errorsystematic error.

poorly timed actions

instrument parallax error

It is usually possible to eliminate systematic errors before beginning an experiment.
Random Error -» Error due to variations in the performance of an instrument or its operator. Random error exists even when you eliminate systematic error.

(ex) temperature variations

misreading examples of sources of random

not collecting enough dataerror.

variations in the thickness of a surface

Question: Are these examples of systematic error or random error?

PH probe was not calibrated.
You measure the distance travelled by a car, but don’t read the metre stick correctly
You measure the time it takes to drop a ball using a stop watch, but you’re never very accurate.

Answer: A. systematic B. random C. systematic

1.2.7 Precision and Accuracy

When doing an experiment, it is important to show how much error and uncertainty there was. The higher the accuracy and precision of an investigation, the lower the uncertainty.
Accuracy -» How close a measurement is to the accepted value, indicated by relative or percentage error in the measurement. An accurate experiment has a low systematic error.
Precision -» An indication of the agreement among a number of measurements made in the same way, indicated by the absolute error. A precise experiment has low random error.

(ex 1) Shooting a duck:

(ex 2) Making a measurement:

Experiment: Throwing darts at a dartboard

1.2.8 Reducing Random Error

Doing many measurements can make it easier to identify the random errors in an experiment. It can also reduce the number of random errors.
Repeating measurements many times will not reduce systematic errors.
Using more precise instruments can reduce random errors.
Controlling the different variables can reduce random errors.

1.2.9 Significant Figures

When using an instrument (a ruler, for example) there is a certain degree of precision.

(ex)

How long is the worm? Using the ruler, we can see that it is at least 4 cm. Looking more closely, we can see that the length is maybe 4.3 or 4.4 cm. We have to guess. Let’s say 4.3 cm.

So, the measurement is

4.3 cm

CertainUncertain

In a measurement, the total number of digits that are known for certain, plus the first uncertain digit, are known as the significant figures. So, the measurement in the example above has two significant figures.
Significant Figures -» The digits in a measurement that are known with certainty followed by the first digit which is uncertain.
Rules for significant figures:

All non-zero digits are significant. (22.2 has 3 sf)

All zeros between two non-zero digits are significant.(1007 has 4 sf).

For numbers less than one, zeros directly after thedecimal point are not significant. (0.0024 has 2 sf)

A zero to the right of a decimal and following anon-zero digit is significant. (0.0500 has 3 sf)

All other zeros are not significant. (500 has 1 sf)

Scientific notation allows you to give a zerosignificance.

For example, 10 has 1 sf but 1.00 × 101 has 3sf.

When adding and subtracting a series ofmeasurements, the least accurate place value in theanswer can only be stated to the same number ofsignificant figures as the measurement of the serieswith the least number of decimal places.

For example, if you add 24.2 g and 0.51 g and7.134 g, your answer is 31.844 g which has increasedin significant digits. The least accurate place valuein the series of measurements is 24.2 g with onlyone number to the right of the decimal point. Sothe answer can only be expressed to 3sf. Therefore,the answer is 31.8 g or 3.18 × 101 g.

PHYSICAL MEASUREMENTCORE

When multiplying and dividing a series ofmeasurements, the number of significant figures inthe answer should be equal to the least number ofsignificant figures in any of the data of the series.

For example, if you multiply 3.22 cm by 12.34 cm by1.8 cm to find the volume of a piece of wood yourinitial answer is 71.52264 cm3. However, the leastsignificant measurement is 1.8 cm with 2 sf. Therefore,the correct answer is 72 cm3 or 7.2 × 101 cm3.

When rounding off a number, if the digit followingthe required rounding off digit is 4 or less, youmaintain the last reportable digit and if it is sixor more you increase the last reportable digit byone. If it is a five followed by more digits exceptan immediate zero, increase the last reportabledigit. If there is only a five with no digits following,increase reportable odd digits by one and maintainreportable even digits.

For example if you are asked to round off thefollowing numbers to two significant numbers

6.42 becomes 6.4

6.46 becomes 6.5

6.451 becomes 6.5

6.498 becomes 6.5

6.55 becomes 6.6

6.45 becomes 6.4

As a general rule, round off in the final step of a series ofcalculations.

HOMEWORK: Physics 3rd edition pg 9-10, # 1 – 12