CfE Higher Physics: Uncertainties

Lochgelly High School

ContentsUnits, prefixes and scientific notation:............................................................................................................................2

Significant figures:...........................................................................................................................................................2

Scientific notation:..........................................................................................................................................................2

Uncertainties:..................................................................................................................................................................3

Random uncertainty:...................................................................................................................................................3

Percentage uncertainty [%Δ].......................................................................................................................................4

Systematic Uncertainty...............................................................................................................................................5

Data Analysis...................................................................................................................................................................6

Individual uncertainties...............................................................................................................................................6

Final analysis for overall result........................................................................................................................................7

Problems.........................................................................................................................................................................8

Solutions........................................................................................................................................................................13

1

Units, prefixes and scientific notation:The units used in the physics course are the standard SI units that you have been using in the past.

examples:displacement s metres mvoltage V volts Venergy E joules J

You will also be expected to remember the prefixes shown below for values in physics.

prefix symbol multiplying factorp Pico x10-12

n Nano x10-9

μ micro x10-6

m milli x10-3

k Kilo x103

M mega x106

G Giga x109

T Tera x1012

Significant figures:In physics the correct use of significant figures is important. This is not just for aesthetic reasons, it has a practical implication.

For example, Jimmy runs 10.0m in the corridor in 3.72 seconds.

If we calculate Jimmy’s speed we obtain an answer of 2.688172043 m/s.

If we give this as our final answer we are effectively saying that we can measure this speed accurately to 10-9m/s. This is an unreasonable level of accuracy for a school laboratory.

The measurements made in order to conduct the experiment were given to 3 sig. figs. so the final value should also be given to this level of accuracy.

So the final value for Jimmy’s average speed should be 2.69m/s.

Scientific notation:You should be familiar with the use of scientific notation in physics. It will be very important for you to use scientific notation in this course since we will be dealing with the physics of the very small ~ 10 -15, as well as the physics of the very large ~ 1015. If you don’t use scientific notation for these sorts of values then your notebook is going to be filled with zeros 000 000 000 000 000 000 000 000 000 000

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Uncertainties:Any measurement made will include an uncertainty in the value obtained. This uncertainty should be taken into account when analysing data.

We will concern ourselves with two main types of uncertainty involved in this course.

Scale Reading Uncertainty This value indicates how well an instrument scale can be read.

An estimate of reading uncertainty for an analogue scale is generally taken as:

± half the least division of the scale.

For a digital scale it is taken as

± 1 in the least significant digit displayed.

Examples

8.94 s

(8.94 + 0.01) s

Time lies between (8.93 and 8.95) s.

6 7

(6.60 + 0.05) cm

Length lies between (6.55 and 6.65) cm

cm0 10

m

(9.0 + 0.5) m

Length lies between (8.5 and 9.5) m

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Random uncertainty:

This is an uncertainty that can be analysed when measurements are repeated. The measurements must be of the same quantity. For example a ball is dropped from a height of 5m and the time for it to fall to the ground is measured using a stop clock.

The measured time should be the same for each drop. In practice the readings are unlikely to be the same; the person measuring the time may not stop the timer at the exact point of impact. In this situation the time is as likely to be too long as too short.

It is possible to take some account of this variation by calculating the mean value of the results.

We also want to have some indication as to how reliable the result obtained is. This can be done by calculating a numerical value for random uncertainty.

random uncertaintity=max . value -min . valuenumber of measurements

Let us examine two different sets of experimental results.

measurement set A set B1 1.00 1.132 0.99 1.023 1.00 0.854 1.03 0.975 1.03 1.08

Set A: mean = (1.01+0.99+1.00+1.02+1.03)/5= 1.01

Set B: mean = (1.13+1.02+0.85+0.97+1.08)/5= 1.01

Both sets of results give the same mean value, but are both sets of results equally reliable?

This can be determined by calculating the random uncertainty for each set.

Set A: random unc .[ Δt ] = 1 .03 -0 . 995

=0 . 008=0 .01

Set B: random unc .[ Δt ] = 1 . 13 -0.855

=0 .056=0 .06

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Set A (1.01 ± 0.01)s

Set B (1.01 ± 0.06)s

From these results it is clear that set A is more reliable than set B.

If we wish to see how much more reliable one set of values is compared to another then we must calculate the percentage uncertainty in the values. This is definitely the case where we have different quantities to compare. For example the mass of the ball was measured before each drop and a further set C results were obtained.

Set C (0.58 ± 0.02)kg

It is not immediately obvious the order of reliability of the sets of results.

Percentage uncertainty [% ]Δ

Uncertainties can also be given as a percentage of the measured value:

%Δ=random uncertaintymean value

x100 %

Using the example above:

%Δ set A = 0 .011. 01

x100%=1%

%Δ set B = 0 .061 .01

x100 %=6 %

%Δ set C = 0 .020 .58

x 100 %=4 %

This shows that set C is more reliable than set B, but less reliable than set A.

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Systematic UncertaintyThis type of uncertainty tends to be due to either the measuring instrument or the method of measurement itself.

For example look at the ruler shown in diag. 1The zero on the scale is not at the edge of the ruler.

A student uses the ruler to measure the height of a clamp stand arm in diag. 2.

If they do not take the offset into account then the measurement will be out by around 6mm. Using the ruler in this fashion will give every length measurement incorrect with the same 6mm offset.

This is a situation that could be rectified easily by applying a 6mm correction factor to each measurement, but it does illustrate the nature of the uncertainty. It may be that you are using a wooden metre stick that has shrunk over the years; it would not be obvious that the scale was incorrect.

Systematic uncertainties can often show themselves in the analysis of data. For instance a graph is plotted of a relationship that would be expected to pass through the origin, but in fact cuts either the x- or y- axis.

The shift of the graph could be due to some systematic uncertainty in the measurements leading to the graph.

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possible systematic effect

NOT

XX

XXXX

X

XX

XXXX

X

This is why you should never force a line through the origin if the data points do not support it.

Comparison of Uncertainties

When comparing uncertainties, it is important to take the percentage in each.

Suppose in an experiment the following uncertainties were found.

Systematic = 0.1 %

Scale Reading = 2 %

Random = 0.5 %

The overall uncertainty should be taken as the highest percentage uncertainty.

In this case, this would be the reading uncertainty at 2 %.

Note: since accuracy is now being quantified, it is essential when using a calculator that all the figures are not taken down, since every number stated indicates the degree of accuracy.

As a general rule, your answer should contain the same number of significant figures as the least accurate reading.

Data AnalysisA student conducts an experiment to measure the average speed of a trolley running down a slope between two fixed marks.

The student measures the distance between the marks and the time for the trolley to move between the marks. This process is repeated four more times.

A set of experimental results is shown below.

measurement 1 2 3 4 5 meandistance(m) 0.85 0.84 0.85 0.83 0.86 0.85

time(s) 1.12 1.24 1.19 1.04 1.22 1.16The average speed is calculated as normal

average speed=distance (mean )time (mean )

=0 . 851 .16

=0 .73ms−1

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d

The reliability of the experiment can be determined by calculating the uncertainty in the final result. To do this we need to calculate the uncertainty in the individual measurements.

Individual uncertainties

Δ distance= 0 .86 - 0 .835

=0.006=0 .1m

Δ time=1. 24 - 1 .045

=0 .04 s

These numbers cannot be easily compared since they relate to different quantities and measurements. The percentage uncertainty should be calculated.

%Δ distance=0 . 010 .85

x100 %=1 .2%

%Δ time=0 .041 .12

x 100 %=3.6%

The %Δ in the time measurement is larger than that for the distance. It is three times larger. In this case it is appropriate for us to find the largest %Δ and use this as the overall %Δ.

In any situation you come up against in Higher questions there will always be one quantity that has a significantly high %uncertainty. It is this %uncertainty that should be used in your final result.

There are methods that would allow you to combine uncertainties of measurements where the %uncertainties have comparable values, however this analysis is beyond the scope of the Higher course.

Final analysis for overall result

average speed = 0.73m/s ± 3.6%

3.6% of 0.73 = 0.026 = 0.03

so average speed = (0.73 ± 0.03) m/s

The final result should always be written in this form.

(final value ± uncertainty)

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Problems

1. State the scale reading uncertainty in the following voltmeter reading.

a. + 0.25 Vb. + 0.5 Vc. + 1.0 Vd. + 2.0 Ve. + 5.5 V

2. A student carries out an investigation to measure the time taken for ten complete swings of a pendulum. The following values are obtained for the time for ten complete swings.

What is the random uncertainty in these results?

a. + 0.01sb. + 0.02sc. + 0.1sd. + 0.2se. + 1.0s

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3. A student carries out three investigations into the variation of voltage and current. The results obtained are shown in the Graphs A, B and C.

Graph A

Graph B

Graph C

Which of the following statements is/are true?• I Graph A shows a systematic uncertainty• II Graph B shows a proportional relationship• III Graph C shows a systematic uncertainty

a. I onlyb. II onlyc. I and II onlyd. I and III onlye. I, II and III

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4. In an experiment the following measurements and uncertainties are recorded.

The measurement which has the largest percentage uncertainty is the:a. Temperature riseb. Heater currentc. Heater voltaged. Timee. Mass of liquid

5. In an investigation the acceleration of a trolley down a slope is found to be 2.5 ms-2 ± 4%.The absolute uncertainty in this value of acceleration is:

a. ± 0.04 m s-2

b. ± 0.1 m s-2

c. ± 0.4 m s-2

d. ± 1.0 m s-2

e. ± 4.0 m s-2

6. In an investigation the voltage across a resistor is measured as 20 V ± 2 V and the current through it is 5.0 A ± 0.1 A. The percentage uncertainty in the power is:

a. 0.1%b. 2%c. 3%d. 10%e. 12%

7. Specific heat capacity can be found from the experimental results given below.Which one of the following measurements creates most uncertainty in the calculated value of the specific heat capacity?

a. Power = 2000 ± 10 Wb. Time = 300 ± 1 sc. Mass = 5.0 ± 0.2 kgd. Final temperature = 50 ± 0.5oCe. Change in temperature = 30 ± 1oC

8. The light coming from a spectral lamp is investigated and the following data is obtained.λ = 450 nm ± 10%

f = 6.7 x 1014Hz ± 2%This data is used to estimate the speed of light. The absolute uncertainty in this estimateof the speed of light is:

a. ± 2.0 m s-1

b. ± 10 m s-1

c. ± 6.0 x 106 m s-1

d. ± 3.0 x 107 m s-1

e. ± 3.6 x 107 m s-1

9. Two forces P and Q act on an object X as shown.

The value of the unbalanced force acting on the object X and the percentage uncertainty in this value, expressed in the form value ± absolute uncertainty is:

a. 14.48 N ± 0.03Nb. 14.48 N ± 0.08Nc. 14.48 N ± 0.5N

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d. 18.22 N ± 0.03 Ne. 18.22 N ± 0.08N

10. Describe clearly the difference between systematic uncertainty and random uncertainty.

11. The ammeter and voltmeter shown are used to calculate a value for resistance.Calculate a value for the resistance and estimate the percentage uncertainty in the value.

12. The density of a liquid is calculated using the formula:

density= massvolume

If the mass is 62.7 ± 0.1g and the volume is 50 ± 1 cm3, estimate the percentage uncertainty in the value of the density.

13. Calculate the percentage uncertainties for the following absolute readings:a. (4.65 ± 0.05) Vb. (892 ± 5) cmc. (1.8 ± 0.4) Ad. (2.87 ± 0.02) se. (13.8 ± 0.5) Hzf. (5.2 ± 0.1) m.

14. Manufacturers of resistors state the uncertainty in their products by using colour codes.Gold - 5 % accuracy. Silver - 10 % accuracy.

Calculate the possible ranges for the following resistors for each colour.a. 1 kΩb. 10 kΩc. 22 kΩ

15. For each of the following scales, write down the reading and estimate the uncertainty.

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0.1

2 4 6

mm

1 2 3

cm 0.2

0.3

Co

a) b) c)

7.84 s 1.005mA

195e) f) g)

16. Calculate the mean time and random uncertainty for the following readings:

0.8s, 0.6 s, 0.5 s, 0.6 s and 0.4 s.

17. A student uses light gates and suitably interfaced computer to measure the acceleration of a trolley as it moves down a slope. The following results were obtained.

a(m s-2) 5.16, 5.24, 5.21, 5.19, 5.12, 5.20, 5.17, 5.19.Calculate the mean acceleration and the corresponding random uncertainty.

18. AB is measured using a metre stick. A trolley is timed between AB.The following results were obtained.AB = (60.0 + 0.1) cmt(s) 1.21, 1.21, 1.26, 1.27, 1.24 and 1.28.

Express the average speed in the form (value ± absolute uncertainty).

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Solutions

Q1: a) ± 0.25 VQ2: d) ± 0.2 sQ3: c) I and II onlyQ4: a) Temperature riseQ5: b) ± 0.1 m s-2Q6: d) 10%Q7: c) Mass = 5.0 ± 0.2 kgQ8: d) ± 3.0 x 107 m s-1Q9: b) 14.48 N ± 0.08NQ10) Q11) 7.1Ω ±3%Q12) 1250kg m-3 ± 2%Q13)

(a) (b) 0.6 %(c) 22 %(d) 0.7 %(e) 3.6 %(f) 1.9 %

Q14.(a) Gold 5 % of 1kΩ = 0.05 kΩ

Silver 10 % of 1kΩ = 0.1kΩ Range is (0.95 - 1.05) kΩRange is (0.9 -1.1) kΩ= (950 - 1050) Ω= (900 - 1100) Ω

(b) Silver (9000 - 11000) Ω Gold (9500 - 10500) Ω(c) Silver (19.8 - 24.2) Ω Gold (20.9 - 23.1) Ω

Q15)(a)( 3.2 ± 0.1) mm (b) (2.30 ± 0.05) cm (c)(0.250 ± 0.005) 0C(d)(2.4 ± 0.2) g (e) (7.84 ± 0.01) s (f) (1.005 ± 0.001) s(g)(195 ± 1) mA

Q16)

t = (0.6 ± 0.1) s

Q17) Mean a = 5.19 m s-2

a = (5.19 ± 0.02) m s-2

Q18) AB = (60.0 ± 0.1) cm = 60 cm ± 0.17 %

Mean t = (1.25 ± 0.01) s = 1.25 s ± 0.8 %

Greatest percentage uncertainty is 0.8 % in the time.

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v = = ± 0.8% = 48.0 cm s-1 ± 0.8% = (48.0 ± 0.4) m s-1

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