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Units of length?

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

The SI 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 metre

- This is the unit of distance. It is the distance traveled by light in a vacuum in a time of 1/299792458 seconds.

The second

- This is the unit of time. A second is the duration of 9192631770 full oscillations of the electromagnetic radiation emitted in a transition between two hyperfine energy levels in the ground state of a caesium-133 atom.

The ampere

- This is the 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 10-7 N on a length of 1 m of the conductors.

The kelvin

- This is the unit of temperature. It is 1/273.16 of the thermodynamic temperature of the triple point of water.

The mole

- One mole of a substance contains as many molecules as there are atoms in 12 g of carbon-12. This special number of molecules is called Avogadro’s number and equals 6.02 x 1023.

The candela (not used in IB)

- This is the unit of luminous intensity. It is the intensity of a source of frequency 5.40 x 1014 Hz emitting 1/683 W per steradian.

The kilogram

- This is the unit of mass. It is the mass of a certain quantity of a platinum-iridium alloy kept at the Bureau International des Poids et Mesures in France.

THE kilogram!

Derived units

Other physical quantities have units that are combinations of the fundamental units.

Speed = distance/time = m.s-1

Acceleration = m.s-2

Force = mass x acceleration = kg.m.s-2 (called a Newton)

(note in IB we write m.s-1 rather than m/s)

Some important derived units (learn these!)

1 N = kg.m.s-2 (F = ma)

1 J = kg.m2.s-2 (W = Force x distance)

1 W = kg.m2.s-3 (Power = energy/time)

Prefixes

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

Prefixes

Power Prefix Symbol Power Prefix Symbol

10-18 atto a 101 deka da

10-15 femto f 102 hecto h

10-12 pico p 103 kilo k

10-9 nano n 106 mega M

10-6 micro μ 109 giga G

10-3 milli m 1012 tera T

10-2 centi c 1015 peta P

10-1 deci d 1018 exa E

Don’t worry! These will all be in the formula book you have for the exam.

PrefixesPower Prefix Symbol Power Prefix Symbol

10-18 atto a 101 deka da

10-15 femto f 102 hecto h

10-12 pico p 103 kilo k

10-9 nano n 106 mega M

10-6 micro μ 109 giga G

10-3 milli m 1012 tera T

10-2 centi c 1015 peta P

10-1 deci d 1018 exa E

Checking equations

If an equation is correct, the units on one side should equal the units on another. We can use base units to help us check.

Checking equations

For example, the period of a pendulum is given by

T = 2π l where l is the length in metres

g and g is the acceleration due to gravity.

In units m = s2 = s

m.s-2

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 nature of the measurement itself.

Individual measurements

When using an analogue scale, the uncertainty is plus or minus half the smallest scale division. (in a best case scenario!)

4.20 ± 0.05 cm

Individual measurements

When using an analogue scale, the uncertainty is plus or minus half the smallest scale division. (in a best case scenario!)

22.0 ± 0.5 V

Individual measurements

When using a digital scale, the uncertainty is plus or minus the smallest unit shown.

19.16 ± 0.01 V

Repeated measurements

When we take repeated measurements and find an average, we can find the uncertainty by finding the difference between the average and the measurement that is furthest from the average.

Repeated measurements - Example

Iker measured the length of 5 supposedly identical tables. He got the following results; 1560 mm, 1565 mm, 1558 mm, 1567 mm , 1558 mm

Average value = 1563 mm

Uncertainty = 1563 – 1558 = 5 mm

Length of table = 1563 ± 5 mm

This means the actual length is anywhere between 1558 and 1568 mm

Precision and Accuracy

The same thing?

Precision

A man’s height was measured several times using a laser device. All the measurements were very similar and the height was found to be

184.34 ± 0.01 cm

This is a precise result (high number of significant figures, small range of measurements)

Accuracy

Height of man = 184.34 ± 0.01cm

This is a precise result, but not accurate (near the “real value”) because the man still had his shoes on.

Accuracy

The man then took his shoes off and his height was measured using a ruler to the nearest centimetre.

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 socks on using the laser device.

Height = 182.23 ± 0.01 cm

This is precise (high number of significant figures) AND accurate (near the real value)

Random errors/uncertainties

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

Systematic/zero errors

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

Systematic/zero errors

This is normally caused by not measuring from zero.

For this reason they are also known as zero errors/uncertainties. Finding an average doesn’t help.

Systematic/zero errors

Systematic errors are sometimes hard to identify and eradicate.

Uncertainties

In the example with the table, we found the length of the table to be 1563 ± 5 mm

We say the absolute uncertainty is 5 mm

The fractional uncertainty is 5/1563 = 0.003

The percentage uncertainty is 5/1563 x 100 = 0.3%

Uncertainties

If the average height of students at BSH is 1.23 ± 0.01 m

We say the absolute uncertainty is 0.01 m

The fractional uncertainty is 0.01/1.23 = 0.008

The percentage uncertainty is 0.01/1.23 x 100 = 0.8%

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.

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 10.0 ± 0.1 cm, width 5.0 ± 0.1 cm and height 6.0 ± 0.1 cm.

Volume = 10.0 x 5.0 x 6.0 = 300 cm3

% uncertainty in length = 0.1/10 x 100 = 1%

% uncertainty in width = 0.1/5 x 100 = 2 %

% uncertainty in height = 0.1/6 x 100 = 1.7 %

Uncertainty in volume = 1% + 2% + 1.7% = 4.7%

(4.7% of 300 = 14)

Volume = 300 ± 14 cm3

This means the actual volume could be anywhere between 286 and 314 cm3

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?

Difference = 44 ± 2 cm

Who’s going to win?

New York Times

Latest opinion poll

Bush 48%

Gore 52%

Gore will win!

Uncertainty = ± 5%

Who’s going to win?

New York Times

Latest opinion poll

Bush 48%

Gore 52%

Gore will win!

Uncertainty = ± 5%

Who’s going to win?

New York Times

Latest opinion poll

Bush 48%

Gore 52%

Gore will win!

Uncertainty = ± 5%

Uncertainty = ± 5%

Who’s going to win

Bush = 48 ± 5 % = between 43 and 53 %

Gore = 52 ± 5 % = between 47 and 57 %

We can’t say!

(If the uncertainty is greater than the difference)

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