Errors and uncertainties

Plot the missing data point on the graph above and draw a line of best fit.

Describe the relationship suggested by the graph and suggest an explanation.

Calculate the magnitude of the recession velocity of M87.

velocity =........ km s^–1

Estimate the distance in light years to M87 using Hubble’s law.

Hubbles constant H_o = 2.5 × 10^–19 s^–1

Estimate the percentage uncertainty in the distance calculated in part (b) using the width of the emission line in the diagram.

percentage uncertainty = +/–........ %

Other methods have shown M87 is actually 50 million light years from Earth.

Suggest why M87 might contradict Hubble’s law.

Suggest one reason why the Big Bang theory is still the accepted model of the origin of the Universe.

A student carries out an experiment to measure the frequency, f, of the mains electricity supply, using the apparatus shown in the diagram.

The wire passes between the poles of a magnet. It is held in tension by the weight W of the mass hanger. The fixed bridge and the movable bridge are separated by distance L.

Mains frequency current is passed through the wire. The student adjusts the movable bridge until the wire vibrates at its fundamental frequency. The weight W and the length L are recorded. This is repeated for several different weights.

Explain how a stationary wave is formed on the wire.

It can be shown that the fundamental frequency f₀ of a wire of length L under tension W is given by the equation

where is the mass per unit length of the wire, and has a value of 8.0 × 10^–4 kg m^–1 ± 2%.

Suggest why the magnetic flux density, B, of the magnet does not appear in this equation.

Suggest a method to determine the mass per unit length, , of the wire.

The graph shows the results of the experiment.

The student draws a best fit line on the graph. The gradient of the best fit line is 2.81 N^1/2m^–1.

Draw a worst acceptable line on the graph.

Use your worst acceptable line to calculate the percentage uncertainty in the gradient of the best fit line.

gradient = 2.81 ±........ N^1/2m^–1%

Calculate a value for the mains frequency, f, including a percentage uncertainty.

f =........ Hz ±........ %

Evaluate the accuracy of your value for f, given that the true value is 50.0 Hz.

An electroscope is an instrument used to measure electric charge.

The diagram shows one type of electroscope.

The diagram below shows the needle of the electroscope when it is negatively charged. The centre of mass, X, of the needle has been marked on the diagram.

Suggest why the centre of mass of the needle is a short distance below the pivot.

A student charges the electroscope by connecting the metal plate to the negative terminal of a high voltage power supply.

They record the deflection angle of the needle for a range of values of the voltage measured using a voltmeter.

Their results are shown in the graph below.

The student concludes that the voltmeter has a zero error of 300 V.

Explain their reasoning.

The capacitance of the electroscope is 20 pF.

Estimate the number of electrons stored on the electroscope when the deflection angle is 45°.

Thermistors are circuit components whose resistance varies with temperature.

There are two major types; negative temperature coefficient (NTC) thermistors, whose resistance decreases with increasing temperature and positive temperature coefficient (PTC) thermistors, whose resistance increases with increasing temperature.

A student is investigating how the resistance of a thermistor varies with temperature by measuring current and voltage. The thermistor is placed in a water bath and the temperature of the water measured using a thermometer.

The diagram below shows how the student set up the experiment (water bath not shown). The circuit has been set up incorrectly.

Describe how the student should change the circuit.

The circuit was corrected and then used to collect data.

The table shows data collected from the investigation.

Temperature / C	Current / mA	Voltage / V
30	0.75	3.00
40	1.10	3.00
50	1.51	3.00
60	2.10	3.00
70	2.80	3.00
80	3.66	3.00
90	4.76	3.00

The axes below show a plot of current against temperature. The first four points from the table have been plotted. Plot the remaining points.

Draw a suitable line of best fit through the data points.

Describe, using the graph and calculations using data from the table, how the resistance of the thermistor varies for increasing temperature. Hence determine whether the thermistor the student used was an NTC or a PTC thermistor.

The thermistor is used in a temperature‑sensing circuit for a heating system to warm milk for a baby.

The student considers two possible designs for the circuit which are shown below.

In each circuit, the voltage V_out across the thermistor is connected to the heating system for warming the milk.

Discuss which circuit may be suitable for the heating system by considering the response of the circuit to changes in temperature.

A group of students investigate the circuit shown in the figure below.

There are N capacitors, each of capacitance C, connected in parallel.

There are N resistors, each of resistance R, connected in series.

Initially, the students close the switch X. They then note the reading on the ammeter.

The students then open the switch. They record the time T for the reading on the ammeter to fall to half of its initial value.

The table below shows the students’ results.

Show that T = (ln 2)N²RC. (a)

The results table is repeated below.

Complete the last column for N = 1 and N = 2 in the table below.

The students begin to plot a graph of T (y-axis) against N² (x-axis). (d)

Draw a straight line of best fit on the graph.

Calculate the gradient of the straight line of best fit.

The value of C is known to be 1000 F ± 5%.

Use your gradient value from (iii) to find a value for R, in units of k , including an absolute uncertainty.

R =........ ±........ k

Following the investigation, the students discovered that the sixth 1000 F capacitor connected to the circuit was actually two 470 F capacitors connected in parallel.

State the type of error caused by this mistake.

Explain the effect that this error would have had on the calculated value of R.

Explain what is meant by

internal resistance of a cell

e.m.f. of a cell.

18* A student is attempting to determine the value of g, the acceleration due to gravity, by two different methods.

Method 1. Vertical drop method

Measuring the time of fall of a small dense ball being dropped from rest from different heights.

Method 2. Rolling ball method

Measuring the time it takes for the same ball to roll 1.900 m down a ramp, set at different angles.

Single sets of results are shown below. The times were measured using a standard stopwatch operated by the student.

Compare and discuss the uncertainties of the two values of g that could be obtained using these single measurements.

Describe how the student would analyse both sets of data when a full range of results has been taken.

Additional space if required

Show that the equation above is homogeneous by reducing the equation to SI base units.

Explain why the magnitude of the resultant force F on the glider is given by F = 2 kx where x is the displacement at any time.

State and explain the effect, if any, of increasing the initial displacement on the period of the subsequent motion.

Masses are added to the glider, and the measurement of 20 T repeated.

The results table is below.

Describe two different errors in the table.

Plot the first data point from the table on the graph below.

The other points have all been plotted. The table of results is repeated on the opposite page.

Include on your graph a line of best fit.

Use the graph to determine the value of k.

=........ N m^–1 k

When the initial displacement is increased, one spring increases its extension while the extension of the other spring decreases.

Explain why the maximum kinetic energy of the motion increases.

The Hipparcos space telescope used stellar parallax with a precision of 9.7 × 10^–4 arcseconds to determine the distance to stars.

One of the stars studied was Polaris A. Data about this star is in the table below.

Estimate the maximum stellar distance in parsecs that could be measured using Hipparcos.

Calculate the percentage uncertainty in the calculated value of the distance to Polaris A.

A continuous stream of particles called a solar wind flows from the surface of the star into the surrounding space.

These particles include helium nuclei of mass 6.6 × 10^–27 kg.

Assume that the atmosphere is modelled as an ideal gas.

Show that the typical kinetic energy of a helium nucleus in the atmosphere is about 10^–16 J.

The gravitational potential energy of a helium nucleus in the outer layer of the star is –2.3 × 10^–16 J.

Calculate the gravitational potential energy U at the maximum distance from the star that a helium nucleus could reach.

Calculate the distance from the centre of the star reached by this helium nucleus.

Explain why the star has a solar wind that reaches a much greater distance from the star than found in (iii).

A student uses the circuit below to investigate the resistivity of a wire.

The cell has e.m.f. and internal resistance r. The wire has resistivity and diameter d.

The student takes five measurements of the diameter of the wire, which are shown in the table below.

Suggest how the student made these measurements.

The student calculates the value of the diameter as d = 0.455 ± 0.005 mm.

Explain how the student calculated the value of the diameter, and its uncertainty, from the data in the table above.

The student varies the length L of the wire in the circuit and records the current I using the ammeter.

The student plots a graph of against L. The data points, error bars, line of best fit and a line of worst fit are shown in the graph below.

The cell has e.m.f. = 1.45 ± 0.05 V

The wire has diameter d = 0.455 ± 0.005 mm

1 Calculate the gradient of the best fit line and use this to determine a value for the resistivity of the wire.

You are not required to determine an uncertainty.

2 Determine a value for the internal resistance r of the cell and its absolute uncertainty.

r =........ ±........

The diagram below shows two insulated-copper coils A and B connected in circuits.

Both coils are individually wrapped around the same iron rod. Coil A is connected to a cell and a switch. Coil B is connected to a filament lamp.

The switch is initially closed and the lamp is off.

The switch is then opened. The lamp flashes on for a brief time, and then remains off.

Explain these observations in terms of magnetic flux.

A student is carrying out an experiment using a search coil.

A long solenoid is connected to an alternating current supply.

The search coil is placed at one end of the solenoid. The plane of the search coil is tilted such that it makes an angle with the central axis of the solenoid. The maximum alternating induced electromotive force (e.m.f.) across the ends of the search coil is E₀.

Name an instrument that can be used to determine E₀.

The equation for E₀ is:

E₀ = K I₀ANf sin

where I₀ = maximum current in the solenoid, A = cross-sectional area of the search coil, N = number of turns of the search coil, f = frequency of the alternating current in the solenoid and K = 4.0 × 10^–3 V A^–1 m^–2 s.

The magnitude of the induced e.m.f. in the search coil can be determined using Faraday’s law of electromagnetic induction:

e.m.f. = rate of change of magnetic flux linkage

In the experiment, angle is changed and E₀ measured.

Suggest the quantity, or quantities, in the equation E₀ = K I₀ANf sin linked to

1 the ‘rate’ part of the law

2 the ‘change of magnetic flux linkage’ part of the law.

The student plots a straight-line graph of E₀ against sin .

Determine f, including the absolute uncertainty. Write your value of f to 2 significant figures.

I₀ = (8.0 ± 0.2) A

f =........ ±........ Hz

The student decides to plot T against R to obtain a straight-line graph.

Show that the line has gradient = C ln2.

The data points plotted by the student are shown below.

1 Draw a best-fit straight line through the data points and use the gradient of this line to determine C.

2 Use your answer in (ii)1 to calculate the permittivity of the paper.

=........ F m^–1

A student investigates the oscillations of a uniform rod of length L which is pivoted at the top, as shown in Fig. 2.1.

Describe how to determine accurately the period T of oscillations of this rod.

The relationship between the frequency f of the oscillations of the rod and its length L is

where g is the acceleration of free fall.

The student varies the length L of the rod and determines the period T for each length.

The student plots a graph of T² against L, shown in Fig. 2.2. A line of best fit has already been drawn.

Show that the gradient of the graph is given by the equation

Draw a line of worst fit on Fig. 2.2.

Use your line of worst fit to calculate the percentage uncertainty in g.

The diagram below shows the top-view of a long current-carrying wire.

The direction of the current in the wire is into the plane of the paper.

Draw at least three field lines to indicate the magnetic field pattern around this wire.

The diagram below shows a circuit to charge a capacitor.

The electromotive force (e.m.f.) E of the cell is 1.48 V and it has negligible internal resistance. The resistance of the resistor is 120 k and the capacitance of the capacitor is 2000 F. At time t = 0 the capacitor is uncharged. The switch is closed at time t = 0.

Calculate the time t when the potential difference across the capacitor is 1.00 V.

A capacitor of capacitance C is connected across a strip of conductive paper.

The switch is moved from X to Y, and the time t for the potential difference across the capacitor to halve is measured.

The time t is given by the expression

t = (Ck ln2) × L

where k is the resistance of the conductive paper per unit length and L is the length of the conductive paper.

The value of C is 1.2 × 10^–3 F.

In an experiment, L is changed and t measured.

The data points are plotted on a t against L grid as shown below.

Draw a straight line of best fit through the data points, and use the gradient of this line to determine k.

The Planck constant h is an important fundamental constant in quantum physics.

Determine the S.I. base units for h.

A researcher is investigating the de Broglie wavelength of charged particles.

The charged particles are accelerated through a potential difference V. The de Broglie wavelength of these particles is then determined by the researcher.

Each particle has mass m and charge q.

The diagram below shows a circuit containing two capacitors which are both initially uncharged. The battery has e.m.f. E and negligible internal resistance.

The switch S is first moved to position A until the capacitor of capacitance C₀ is fully charged.

The switch S is then moved to position B. The initial charge stored by the capacitor of capacitance C₀ is shared between the two capacitors. The final reading on the voltmeter is V.

A student wants to determine the values of E and C₀ by repeating the experiment above and measuring the potential difference (p.d.) V for a selection of capacitors of capacitance C. The student decides to plot a graph of against C.

The data points, error bars and the line of best fit drawn by the student are shown in the graph below.

The gradient of the line of best fit is 51 V^–1 F^–1. The value of E is 9.1 V.

Determine the value of C₀ in millifarads (mF). Write your answer to 2 significant figures.

Draw on the graph a straight line of worst fit.

Use this line to determine the absolute uncertainty in your value of C₀. Write your answer to an appropriate number of significant figures.

A solid cylindrical glass rod has length 20.0 ± 0.1 cm and diameter 5.00 ± 0.01 mm.

What is the percentage uncertainty in the calculated volume of this rod?

A 0.1%

B 0.2%

C 0.7%

D 0.9%

Your answer

This question is about waves.

The period of a progressive wave can be determined from Fig. 16.1. Add a correct label to

the horizontal axis so that the period can be found.

A progressive wave has wavelength , frequency f and period T.

Show that the speed v of the wave is given by the equation v = f.

Calculate the period T of the light wave.

The rays of light are in phase at the dashed line X.

Use your two answers above to state the phase difference in degrees between the light rays at Y.

Question 16 continues on page 12

State Kirchhoff’s second law and the physical quantity that is conserved according to this law.

On Fig. 18.2 draw the straight line of best fit. Determine the gradient of this line.

gradient =........ A^–1 m^–1

Show that the gradient of the line is , where is the resistivity of the material of the wire, A is the area of cross-section of the wire and E is the e.m.f. of the cell.

The e.m.f. E of the cell is 1.5 V. The diameter of the wire is 0.38 mm.

Use your answer to (i) and the equation given in (ii) to determine .

=........ m

Hydrogen atoms excited in a discharge tube only emit four different discrete wavelengths of visible photons.

Show that the energy of a photon of wavelength 486 nm is 4.09 × 10^–19 J. (b) (i)

Fig. 5.4 shows some of the energy levels of an electron in a hydrogen atom.

Draw an arrow on Fig. 5.4 to show an electron transition which would cause the emission

of a photon of wavelength 486 nm.

Describe the Doppler effect.

Explain how ultrasound is used to measure the speed of blood flow in an artery.

Use Fig. 6.2 to show that the speed of the loudspeaker is 7.5 m s^–1. (i)

The speed of sound in this experiment is 330 m s^–1. (ii)

Calculate the maximum change in frequency f of the sound detected by the microphone.

Hence complete the scale on the y-axis of Fig. 6.2.

Mark with an X on Fig. 6.1 the position of the loudspeaker which corresponds to the

point X on Fig. 6.2.

In their laboratory notes, one student writes about the accuracy of the measurements whereas the other writes about their precision.

Define these terms.

Which is the best value for the elementary charge e in terms of both accuracy and precision?

Your answer

The acoustic impedance Z of a material in the shape of a cube can be determined using the equation

where M is the mass of the material, L is the length of each side of the cube and c is the speed of ultrasound in the material.

The percentage uncertainty in L is 1.2 % and the percentage uncertainty in c is 1.8 %. The percentage uncertainty in M is negligible.

What is the percentage uncertainty in Z ?

A 2.2 %

B 3.0 %

C 4.2 %

D 5.4 %

Your answer

Draw the straight line of best fit on Fig 20.1 and determine the gradient of the line.

gradient =........ V m

Calculate the percentage difference between your value in (ii) and the accepted value of the Planck constant.

Identify the two types of errors shown by the data in Fig. 20.1 and suggest how you could have refined the experiment to reduce or eliminate these errors.

Fig. 20.2 shows a gold-leaf electroscope with a clean zinc plate.

The zinc plate, metal stem and the gold-leaf are given a negative charge by briefly connecting the zinc plate to the negative electrode of a high-voltage supply.

The gold leaf is fully diverged.

The position of the leaf is not affected by intense white light from a table lamp incident on the zinc plate. The gold leaf collapses very quickly when low-intensity ultraviolet radiation from a mercury lamp is incident on the zinc plate.

Explain these observations in terms of photons.

Fig. 4.1 shows an arrangement used by a student to determine the acceleration of free fall.

A trolley is attached to a variable mass m by a string which passes over a pulley.

The mass m is released from rest and falls through a fixed height of 0.600 m accelerating the trolley of mass 0.800 kg. When the mass m hits the floor, the trolley then continues to move at a constant velocity v.

This constant velocity v is determined by measuring the time t for the card of length 0.200 m to pass fully through a light gate connected to a timer.

Frictional forces on the trolley and the falling mass m are negligible.

Show that the relationship between v and m is

where g is the acceleration of free fall.

The student records the information from the experiment in a table. The column headings and just the last row for m = 0.600 kg from this table are shown below.

It is suspected that the card on the trolley did not pass at right angles through the light beam.

Discuss, without doing any calculations, the effect this may have on the experimental value for the acceleration of free fall g.

Which is the best estimate of the area of a rectangular field of length 98 ± 3 m and width 47 ± 2 m?

4600 ± 5 m² A

4600 ± 6 m² B

4600 ± 300 m² C

4606 ± 337 m² D

Your answer

A group of students are conducting an experiment in the laboratory to determine the value of absolute zero by heating a fixed mass of gas. The volume of the gas is kept constant. Fig. 17.1 shows the arrangement used by the students.

The gas is heated using a water bath. The temperature of the water is increased from 5 °C to 70 °C. The temperature of the water bath is assumed to be the same as the temperature of the gas. The pressure p of the gas is measured using a pressure gauge.

The results from the students are shown in a table.

/ °C	p / kPa
5 ± 1	224 ± 3
13 ± 1	231 ± 3
22 ± 1	238 ± 3
35 ± 1	248 ± 3
44 ± 1
53 ± 1	262 ± 3
62 ± 1	269 ± 3
70 ± 1	276 ± 3

Describe and explain how the students may have made accurate measurements of the temperature .

Fig. 17.2 shows the pressure gauge. Measurements of p can be made using the kPa scale or the psi (pounds per square inch) scale. The students used the psi scale to measure pressure and then converted the reading to pressure in kPa.

Suggest why it was sensible to use the psi scale to measure p.

Plot the missing data point and the error bars on Fig. 17.3.

* Explain what is meant by absolute zero. Describe how Fig. 17.3 can be used to determine the value of absolute zero. Determine the value of absolute zero. You may assume that the gas behaves as an ideal gas.

Describe, without doing any calculations, how you could use Fig. 17.3 to determine the actual uncertainty in the value of absolute zero in (c)(ii).

The experiment is repeated as the water bath quickly cools from 70 °C to 5 °C. Absolute zero was found to be −390 °C.

Compare this value with your value from (c)(ii) and explain why the values may differ. Describe an experimental approach that could be taken to avoid systematic error in the determination of absolute zero.

State one S.I. base quantity other than length, mass and time.

Show that the total resistance R of the wires is given by the equation

A student uses the equation in (i) to determine R. The table below shows the data recorded by the student in her lab book.

Quantity	Value
	4.7 10−7 m
L	9.5 ± 0.1 cm
d	0.270 ± 0.003 mm

1. Name the likely instruments used by the student to measure L and d.

2. Use the data in the table and the equation in (i) to determine R and the absolute uncertainty. Write your answer to the correct number of significant figures.

3. The instrument used to measure d has a zero-error. The measured d is much larger than the actual value. Discuss how the actual value of R compares with the value calculated above.

* A student conducts an experiment to confirm that the uniform magnetic flux density B between the poles of a magnet is 30 mT.

A current-carrying wire of length 5.0 cm is placed perpendicular to the magnetic field.

The current I in the wire is changed and the force F experienced by the wire is measured. Fig. 22.1 shows the graph plotted by the student.

The student’s analysis is shown on the graph of Fig. 22.1 and in the space below.

F = BIL

This is just 2 mT out from the 30 mT value given by the manufacturer, so the experiment is very accurate.

Evaluate the information from Fig. 22.1 and the analysis of the data from the experiment. No further calculations are necessary.

Use Faraday’s law of electromagnetic induction to explain why the filament lamp is lit.

The primary coil has 400 turns and the secondary coil has 20 turns. The potential difference across the lamp is 12 V and it dissipates 24 W. The transformer is 100% efficient.

1. Calculate the current in the primary coil.

2. The alternating voltage supply is replaced by a battery and an open switch in series. The switch is closed. The lamp is lit for a short period of time and then remains off. Explain this observation.