Measurements and Uncertainities

IB Grade 11 Physics Lesson 1

The realm of physics

• 1.1.1 State and compare quantities to the nearest order of magnitude.

Throughout the study of physics we deal with a wide range of magnitudes. We will use minuscule values such as the mass of an electron and huge ones such as the mass of the (observable) universe. In order to easily understand the magnitude of these quantities we need a way to express them in a simple form, to do this, we simply write them to the nearest power of ten (rounding up or down as appropriate).That is, instead of writing a number such as 1000, we write 103 .

• The use of orders of magnitude is generally just to get an idea of the scale and differences in scale of values. It is not an accurate representation of a value. For example, if we take 400, it’s order of magnitude is 102 , which when we calculate it gives 10 x 10 = 100. This is four times less than the actual value, but that does not matter. The point of orders of magnitude is to get a sense of the scale of the number, in this case we know the number is within the 100s.

1.1.2 State the ranges of magnitude of distances, masses and times that occur in the universe, from smallest to greatest.

• Distances:• sub-nuclear particles: 10-15 m• extent of the visible universe: 10+25 m• Masses:• mass of electron: 10-30 kg• mass of universe: 10+50 kg• Times:• passage of light across a nucleus: 10-23 s• age of the universe : 10+18 s

1.1.3 State ratios of quantities as differences of orders of magnitude.

• Using orders of magnitude makes it easy to compare quantities, for example, if we want to compare the size of an an atom (10-10 m) to the size of a single proton (10-15 m) we would take the difference between them to obtain the ratio. Here, the difference is of magnitude 105meaning that an atom is 105 or 100000 times bigger than a proton.

1.1.4 Estimate approximate values of everyday quantities to one or two significant figures and/or to

the nearest order of magnitude.

• Significant figuresTo express a value to a certain amount of significant figures means to arrange the value in a way that it contains only a certain amount of digits which contribute to its precision.

• For example, if we were asked to state the value of an equation to three significant figures and we found the result of that value to be 2.5423, we would state it as 2.54.

• Note that 2.54 is accurate to three significant figures as we count both the digits before and after the point.

• The amount of significant figures includes all digits except:

• leading and trailing zeros (such as 0.0024 (2 sig. figures) and 24000 (2 sig. figures)) which serve only as placeholders to indicate the scale of the number.

• extra “artificial” digits produced when calculating to a greater accuracy than that of the original data, or measurements reported to a greater precision than the equipment used to obtain them supports.

Rules for identifying significant figures:All non-zero digits are considered significant (such as 14 (2 sig. figures) and 12.34 (4 sig. figures)).Zeros placed in between two non-zero digits (such as 104 (3 sig. figures) and 1004 (4 sig. figures))Trailing zeros in a number containing a decimal point are significant (such as 2.3400 (5 sig. figures) note that a number 0.00023400 also has 5 sig. figures as the leading zeros are not significant).

Note that a number such as 0.230 and 0.23 are technically the same number, but, the former (0.230) contains three significant figures, which states that it is accurate to three significant figures. On the other hand, the latter (0.23) could represent a number such as 2.31 accurate to only two significant figures. The use of trailing zeros after a decimal point as significant figures is is simply to state that the number is accurate to that degree.

Another thing to note is that some numbers with no decimal point but ending in trailing zeros can cause some confusion.

For example, the number 200, this number contains one significant figure (the digit 2). However, this could be a number that is represented to three significant figures which just happens to end with trailing zeros.

Typically these confusions can be resolved by taking the number in context and if that does not help, one can simply state the degree of significance (for example “200 (2 s.f.)” , means that the two first digits are accurate and the second trailing zero is just a place holder.

Expressing significant figures as orders of magnitude:To represent a number using only the significant digits can easily be done by expressing it’s order of magnitude. This removes all leading and trailing zeros which are not significant.For example:0.00034 contains two significant figures (34) and fours leading zeros in order to show the magnitude. This can be represented so that it is easier to read as such: 3.4 x 10-4 .

Note that we simply removed the leading zeros and multiplied the number we got by 10 to the power of negative the amount of leading zeros (in this case 4). The negative sign in the power shows that the zeros are leading.A number such as 34000 (2 s.f.) would be represented as 34 x 103 .

Again, we simply take out the trailing zeros, and multiply the number by 10 to the power of the number of zeros (3 in this case).There are a couple of cases in which you need to be careful:A number such as 0.003400 would be represented as 3.400 x 10-3 . Remember, trailing zeros after a decimal point are significant.56000 (3 s.f.) would be represented as 560 x 102 . This is because it is stated that the number is accurate the 3 significant zeros, therefor the first trailing zero is significant and must be included. Note that this can be used as another way to express a value such as 56000 to three significant figures (as opposed to writing “56000 (3 s.f.)”).

RoundingWhen working with significant figures you will often have to round numbers in order to express them to the appropriate amount of significant figures.For example:State 2.342 to three significant figures: would be written 2.34.When representing the number 2.342 to three significant figures we rounded it down to 2.34. This means that when we removed the excess digit, it was not high enough to affect the last digit that we kept.

Whether to round up or down is a simple decision:If the first digit in the excess being cut off is lower than 5 we do not change the last digit which we are keeping. If the first digit in the excess which is being cut off is 5 or higher we increment the last digit that we are keeping (and the rest of the number if required).

For example:State 5.396 to three significant figures: would be written 5.40. This is because we remove the last digit (6) in order to have three significant digits. However, 6 is large enough for it to affect the digit before it, therefor we increment the last digit of the number we are keeping by 1. Note that the last digit of the number that we are keeping is 9, therefor incrementing it by 1 gives 10. Thus we also need to increment the second to last digit by 1. This gives 5.40 which is accurate to three significant figures.

You might think that if a number such as 5.4349 were to be rounded to 3 s.f. it would give 5.44 as the last digit is 9 which is large enough to affect the previous digit which would then become 5. Now that 5 would be large enough to affect the last digit of the number we are keeping which would become 4 (thus 5.44). However, this is not the case, when rounding, we only look at the digit immediately after the one we are rounding to, whether or not that digit would be affected by the one after it is not taken into account. Therefor, the correct result of this question would be 5.43.

1.2.1 State the fundamental units in the SI system.

Many different types of measurements are made in physics. In order to provide a clear and concise set of data, a specific system of units isused across all sciences. This system is called the International System of Units (SI from the French "Système International d'unités").The SI system is composed of seven fundamental units:

Quantity Unit name Unit symbol

mass kilogram kg

time second s

length meter m

temperature kelvin K

Electric current ampere A

Amount of substance mole mol

Luminous intensity candela cd

Note that the last unit, candela, is not used in the IB diploma program.

Figure 1.2.1 - The fundamental SI units

1.2.2 Distinguish between fundamental and derived units and give examples of derived units.

• In order to express certain quantities we combine the SI base units to form new ones. For example, if we wanted to express a quantity of speed which is distance/time we write m/s (or, more correctly m s-1). For some quantities, we combine the same unit twice or more, for example, to measure area which is length x width we write m2.

• Certain combinations or SI units can be rather long and hard to read, for this reason, some of these combinations have been given a new unit and symbol in order to simplify the reading of data.For example: power, which is the rate of using energy, is written as kg m2 s-3. This combination is used so often that a new unit has been derived from it called the watt (symbol: W).

Table 1.2.2 - SI derived unitsSI derived unit Symbol SI base unit Alternative unitnewton N kg m s-2 -joule J kg m2 s-2 N mhertz Hz s-1 -watt W kg m2 s-3 J s-1

volt V kg m2 s-3 A-1 W A-1

ohm Ω kg m2 s-3 A-2 V A-1

pascal Pa kg m-1 s-2N m-2

Below is a table containing some of the SI derived units you will often encounter: 

1.2.3 Convert between different units of quantities.

• Often, we need to convert between different units. For example, if we were trying to calculate the cost of heating a litre of water we would need to convert between joules (J) and kilowatt hours (kW h), as the energy required to heat water is given in joules and the cost of the electricity used to heat the water is a certain price per kW h.

• If we look at table 1.2.2, we can see that one watt is equal to a joule per second. This makes it easy to convert from joules to watt hours: there are 60 second in a minutes and 60 minutes in an hour, therefor, 1 W h = 60 x 60 J, and one kW h = 1 W h / 1000 (the k in kW h being a prefix standing for kilo which is 1000).

1.2.4 State units in the accepted SI format.

• There are several ways to write most derived units. For example: meters per second can be written as m/s or m s-1. It is important to note that only the latter, m s-1, is accepted as a valid format. Therefor, you should always write meters per second (speed) as m s-1 and meters per second per second (acceleration) as m s-2. Note that this applies to all units, not just the two stated above.

1.2.5 State values in scientific notation and in multiples of units with appropriate prefixes.

• When expressing large or small quantities we often use prefixes in front of the unit. For example, instead of writing 10000 V we write 10 kV, where k stands for kilo, which is 1000. We do the same for small quantities such as 1 mV which is equal to 0,001 V, m standing for milli meaning one thousandth (1/1000).

• When expressing the units in words rather than symbols we say 10 kilowatts and 1 milliwatt.

1.2.6 Describe and give examples of random and systematic errors.

• Random errorsA random error, is an error which affects a reading at random.Sources of random errors include:

• The observer being less than perfect• The readability of the equipment• External effects on the observed item• Systematic errors• A systematic error, is an error which occurs at each reading.

Sources of systematic errors include:• The observer being less than perfect in the same way every time• An instrument with a zero offset error• An instrument that is improperly calibrated

• 1.2.7 Distinguish between precision and accuracy.• Precision

A measurement is said to be accurate if it has little systematic errors.

• AccuracyA measurement is said to be precise if it has little random errors.

• A measurement can be of great precision but be inaccurate (for example, if the instrument used had a zero offset error).

• 1.2.8 Explain how the effects of random errors may be reduced.

• The effect of random errors on a set of data can be reduced by repeating readings. On the other hand, because systematic errors occur at each reading, repeating readings does not reduce their affect on the data.

1.2.9 Calculate quantities and results of calculations to the appropriate number of significant figures.

• The number of significant figures in a result should mirror the precision of the input data. That is to say, when dividing and multiplying, the number of significant figures must not exceed that of the least precise value.

• Example:Find the speed of a car that travels 11.21 meters in 1.23 seconds.

• 11.21 x 1.13 = 13.7883• The answer contains 6 significant figures. However, since the

value for time (1.23 s) is only 3 s.f. we write the answer as 13.7 m s-1.

• The number of significant figures in any answer should reflect the number of significant figures in the given data.

1.2.10 State uncertainties as absolute, fractional and percentage uncertainties.

• Absolute uncertaintiesWhen marking the absolute uncertainty in a piece of data, we simply add ± 1 of the smallest significant figure.

• Example:• 13.21 m ± 0.01

0.002 g ± 0.0011.2 s ± 0.112 V ± 1

• Fractional uncertaintiesTo calculate the fractional uncertainty of a piece of data we simply divide the uncertainty by the value of the data.

• Example:• 1.2 s ± 0.1

Fractional uncertainty:

0.1 / 1.2 = 0.0625

• Percentage uncertaintiesTo calculate the percentage uncertainty of a piece of data we simply multiply the fractional uncertainty by 100.

• Example:• 1.2 s ± 0.1• Percentage uncertainty:• 0.1 / 1.2 x 100 = 6.25 %

1.2.11 Determine the uncertainties in results.Simply displaying the uncertainty in data is not enough, we need to include it in any calculations we do with the data.Addition and subtractionWhen performing additions and subtractions we simply need to add together the absolute uncertainties.Example:Add the values 1.2 ± 0.1, 12.01 ± 0.01, 7.21 ± 0.011.2 + 12.01 + 7.21 = 20.420.1 + 0.01 + 0.01 = 0.1220.42 ± 0.12

Multiplication, division and powersWhen performing multiplications and divisions, or, dealing with powers, we simply add together the percentage uncertainties.Example:Multiply the values 1.2 ± 0.1, 12.01 ± 0.011.2 x 12.01 = 140.1 / 1.2 x 100 = 8.33 %0.01 / 12.01 X 100 = 0.083%8.33 + 0.083 = 8.413 %14 ± 8.413 %

Other functionsFor other functions, such as trigonometric ones, we calculate the mean, highest and lowest value to determine the uncertainty range. To do this, we calculate a result using the given values as normal, with added error margin and subtracted error margin. We then check the difference between the best value and the ones with added and subtracted error margin and use the largest difference as the error margin in the result.Example:Calculate the area of a field if it's length is 12 ± 1 m and width is 7 ± 0.2 m.Best value for area:12 x 7 = 84 m2

Highest value for area:13 x 7.2 = 93.6 m2

Lowest value for area:11 x 6.8 = 74.8 m2

If we round the values we get an area of:84 ± 10 m2

1.2.12 Identify uncertainties as error bars in graphs.

When representing data as a graph, we represent uncertainty in the data points by adding error bars. We can see the uncertainty range by checking the length of the error bars in each direction. Error bars can be seen in figure 1.2.1 below:

Figure 1.2.1 - A graph with error bars

1.2.13 State random uncertainty as an uncertainty range (±) and represent it graphically as an "error bar".

In IB physics, error bars only need to be used when the uncertainty in one or both of the plotted quantities are significant. Error bars are not required for trigonometric and logarithmic functions.To add error bars to a point on a graph, we simply take the uncertainty range (expressed as "± value" in the data) and draw lines of a corresponding size above and below or on each side of the point depending on the axis the value corresponds to.

Example:Plot the following data onto a graph taking into account the uncertainty.

Time ± 0.2 s Distance ± 2 m

3.4 13

5.1 36

7 64

Figure 1.2.2 - Distance vs. time graph with error bars

In practice, plotting each point with its specific error bars can be time consuming as we would need to calculate the uncertainty range for each point. Therefor, we often skip certain points and only add error bars to specific ones. We can use the list of rules below to save time:Add error bars only to the first and last pointsOnly add error bars to the point with the worst uncertaintyAdd error bars to all points but use the uncertainty of the worst pointOnly add error bars to the axis with the worst uncertainty

GradientTo calculate the uncertainty in the gradient, we simply add error bars to the first and last point, and then draw a straight line passing through the lowest error bar of the one points and the highest in the other and vice versa. This gives two lines, one with the steepest possible gradient and one with the shallowest, we then calculate the gradient of each line and compare it to the best value. This is demonstrated in figure 1.2.3 below:

InterceptTo calculate the uncertainty in the intercept, we do the same thing as when calculating the uncertainty in gradient. This time however, we check the lowest, highest and best value for the intercept. This is demonstrated in figure 1.2.4 below:

Figure 1.2.4 - Intercept uncertainty in a graphNote that in the two figures above the error bars have been exaggerated to improve readability.

Prepared by Mesut MızrakIBDP Physics Teacher2014