Units of length

Mile, furlong, fathom, yard, feet, inches, Angstroms, nautical miles, cubits

The SI (Système Internationale) system of units There are seven fundamental base units which are clearly defined and on which all other derived units are based: You need to know these

The meter (m) Unit of length A meter is the distance traveled by light in a vacuum in a time of 1/ seconds.

The kilogram (kg) Unit of mass THE kilogram! A kilogram is the mass of a certain quantity of a platinum- iridium alloy kept at the Bureau International des Poids et Mesures in France.

The second (s) Unit of time A second is the duration of full oscillations of the electromagnetic radiation emitted in a transition between two hyperfine energy levels in the ground state of a caesium-133 atom.

If you ever forget what your fundamental units of length, mass, and time are, just think: MK S

The kelvin (K) This is the unit of temperature. It is 1/ of the thermodynamic temperature of the triple point of water.

The ampere (A) Unit of electrical current. It is defined as that current which, when flowing in two parallel conductors 1 m apart, produces a force of 2 x N on a length of 1 m of the conductors.

The mole (mol) Unit of amount of substance (number of atoms/molecules) Defined as number of atoms in 12 g of carbon-12. This special number of atoms is called Avogadro’s number and equals 6.02 x 10 23

The candela (cd) Unit of luminous intensity It is the intensity of a source of frequency 5.40 x Hertz emitting 1/683 Watts per steradian.

Derived units Other physical quantities are derived from the fundamental units. Speed = Density = Volume =length x length x length =m3m3 mass/volume =kg/m 3 or kg m -3 IB uses this notation--Sorry. distance/time =m/sm/sor m s -1

Prefixes It is sometimes useful to express units that are related to the basic ones by powers of ten

Prefixes PowerPrefixSymbolPowerPrefixSymbol teraT10 -2 centic 10 9 gigaG millim 10 6 megaM microμ 10 3 kilok10 -9 nanon picop femtof

Examples 3.3 kilometers = 545 milliseconds = 3.3 x 10 3 meters =3,300 m 545 x seconds =0.545 s

Conversions -You will often need to convert between different units. -This is done using the principle that anything divided by itself equals: =1 x x = 1 60 seconds 1 minute =1

Conversions -We will also use the fact that anything multiplied by 1 equals: ITSELF 8 x 1 = 8 y x 1 =y1 minute x 1 = 1 minute

Conversions -Convert 1 minute to seconds 1 minutex 60 seconds = 1 minute Multiply by 1! 1 minute x 60 seconds = 1 minute Anything divided by itself is 1! 60 seconds -Convert 1 year to seconds 1 yearx days 1 year x 24 hours 1 day x 60 min 1 hour x 60 seconds = 1 min 31,557,600 seconds or x 10 7 s

Let’s try some questions! Pages 1-2 Questions 1-5

Errors/Uncertainties

In every measurement (as opposed to simply counting) there is an uncertainty in the measurement. This is sometimes determined by the apparatus you're using, sometimes by the way you do the measurement.

Individual measurements When using an analog scale, the uncertainty is plus or minus the smallest scale division. 4.2 ± 0.1 cm cm

Individual measurements When using an analog scale, there may be instances where, in a best case scenario, your uncertainty could be half the smallest scale division ± 0.5 V For the most part, let's keep it simple 22 ± 1 V Let's call it

Individual measurements When using a digital scale, the uncertainty is plus or minus the smallest unit shown ± 0.01 V

Repeated measurements When we take repeated measurements we can find the uncertainty by maximum measurement - minimum measurement 2

Repeated measurements - Example Somchai measured the length of 5 supposedly identical tables. He got the following results: This means the actual length is anywhere between 1558 and 1568 mm Length of table = 1563 ± 5 mm Uncertainty = 1567 – = 5 mm Average = 1563 mm 1560mm mm mm mm mm = 5 max - min = mm, 1565 mm, 1558 mm, 1567 mm, 1558 mm

Precision and Accuracy The same thing?

Precision A man’s height was measured several times using a laser device. This is a precise result (high number of significant figures, small range of measurements) ± 0.01 cm All the measurements were very similar and the height was found to be

Precise While the result was precise, it was not accurate (near the “real value”) because the man still had his shoes on, but not Accurate

Accuracy The man then took his shoes off and his height was measured using a ruler to the nearest centimeter. Height = 182 ± 1 cm This is accurate (near the real value) but not precise (only 3 significant figures)

Precise and accurate The man’s height was then measured without his shoes on using the laser device. This is precise (high number of significant figures) AND accurate (near the real value) Height = ± 0.01 cm

Random error Some measurements do vary randomly. Some are bigger than the actual/real value, some are smaller. This is called a random error/uncertainty. Finding an average can produce a more reliable result in this case.

Systematic errors Sometimes all measurements are bigger or smaller than they should be. This is called a systematic error/uncertainty.

Systematic errors This is normally caused by not measuring from zero. For example, if you measured your piece of string from the end of the ruler, not from the “0” mark, you have a systematic error. For this reason they are also known as zero errors. Finding an average doesn’t help.

Systematic/zero errors Systematic errors are sometimes hard to identify and eradicate.

Let’s try some questions.

Uncertainties In the example with the table, we found the length of the table to be 1563 ± 5 mm The percentage uncertainty is 5 mm x 100% = 1563 mm The fractional uncertainty is 5 mm = 1563 mm We say the absolute uncertainty is 5 mm %

Uncertainties In the example with the table, we found the length of the table to be 1563 ± 5 mm The percentage uncertainty is The fractional uncertainty is We say the absolute uncertainty is % 5 mm = 5 mm x 100% = 5 mm 1563 mm

Uncertainties If the average height of students at Senn is 1.55 ± 0.08 m, find the absolute, fractional, and percentage uncertainties for this measurement The percentage uncertainty is The fractional uncertainty is The absolute uncertainty is0.08 m 0.08 m = 5% m x 100% = 1.55 m

Combining uncertainties When we find the volume of a block, we have to multiply the length by the width by the height. Because each measurement has an uncertainty, the uncertainty increases when we multiply the measurements together. V = L x W x H

Combining uncertainties When multiplying (or dividing) quantities, to find the resultant uncertainty we have to add the percentage uncertainties of the quantities we are multiplying.

Combining uncertainties Example: A block has a length of 5.0 ± 0.1 cm, width 3.0 ± 0.1 cm and height 6.0 ± 0.1 cm. Find the volume with uncertainty. This means the actual volume could be anywhere between 85 and 95 cm 3 Volume = 90 ± 5 cm 3 6.0% of 90 = Uncertainty in volume = % uncertainty in height = % uncertainty in width = % uncertainty in length = Volume =90 cm cm x 5.0 cm x 6.0 cm = 0.1/5.0 x 100 = 2.0% 3.3 % 0.1/3.0 x 100 = 1.7 %0.1/6.0 x 100 = 6.0%2.0% + 3.3% + 1.7% = x 0.06 =

Combining uncertainties When adding (or subtracting) quantities, to find the resultant uncertainty we have to add the absolute uncertainties of the quantities we are multiplying.

Combining uncertainties One basketball player has a height of 196 ± 1 cm and the other has a height of 152 ± 1 cm. What is the difference in their heights, with uncertainties? 196 cm cm =44 cm Difference = 44 ± 2 cm Uncertainty:2 cm 1 cm + 1 cm =

Who’s going to win? Latest opinion poll Bush 49% Gore 51% Gore will win! Uncertainty = ±3%

Who’s going to win? Latest opinion poll Bush 49% Gore 51% Gore will win! Uncertainty = ± 3%

Who’s going to win? Latest opinion poll Bush 49% Gore 51% Gore will win! Uncertainty = ±3%

Who’s going to win Bush = 49 ± 3 % = between 46 and 52 % Gore = 51 ± 3 % = between 48 and 54 % We can’t say! (If the uncertainty is greater than the difference)

Let’s try some more questions!