Errors and uncertainties

A student investigated the variation of current with applied potential difference (p.d.) for a filament bulb. Under normal working conditions the potential difference across the bulb is 3.0 V.

The circuit diagram shows the circuit the student used.

Comment on the suitability of the student’s circuit.

When the p.d. across the bulb was 3.0 V, the student recorded a value of 1.29 A for the current.

In his conclusion he stated

This will give an accurate value for the working resistance, as the value of current is precise.

Assess the validity of the student’s statement. You do not need to carry out a calculation.

A student uses a digital balance. She sets the balance to zero and then places a prism on the balance, as shown.

A uniform metre rule is pivoted at its 5.0 cm mark. The metre rule is then supported horizontally at the 30.0 cm mark by the prism, as shown below.

The metre rule is in equilibrium.

Explain why the increase in mass recorded by the balance is not equal to the mass of the metre rule.

Calculate the weight of the metre rule. You should take moments about the pivot.

The student wants to reduce the percentage uncertainty in the value obtained for the weight of the metre rule.

The student repeated her determination of the weight of the metre rule. She moved the prism towards the centre of gravity of the metre rule before taking moments about the pivot.

Deduce whether the percentage uncertainty in the value obtained for the weight of the metre rule is reduced by the student moving the prism. No calculations are required.

Two students, A and B, used a cobalt‑60 source to investigate the inverse square law for a gamma radiation source. Cobalt‑60 is a pure gamma emitter.

The cobalt‑60 source was placed in a source holder. A Geiger‑Müller (GM) tube was connected to a counter and placed a short distance d away.

Student A set up the apparatus as shown.

Student B used the same distance d but rotated the GM tube by 90°, as shown below.

Both students recorded the count for 2 minutes.

Deduce which set‑up will lead to a smaller percentage uncertainty in the count rate.

Explain two modifications to the experimental method which would improve the accuracy of the data.

Student A suggests that the investigation into the inverse square law for a gamma radiation source could be carried out successfully using a radium‑226 source.

Radium‑226 emits alpha and beta radiation as well as gamma radiation.

Explain one modification that would be necessary if a radium‑226 source were to be used.

Fuse wire is made of a metal that has a low melting point. The wire melts when the current exceeds a maximum value.

A student investigated how the cross‑sectional area affected the current at which a wire melted. She used wires of the same length and metal, but different diameters.

The student steadily increased the current through each wire and recorded the maximum current through the wire before it melted.

Describe how the student could determine an accurate value for the cross‑sectional area of a wire.

Explain why a current in a wire causes the resistance of the wire to change.

The length of a wire does not affect the value of current at which the wire melts.

Explain why. Assume the wire behaves as a black body.

A student has 10 identical flat metal discs. One of the discs is shown. Each disc has a diameter d of about 5 cm and a thickness t of about 2 mm.

The student determined the density of the metal the discs are made from.

She used a half‑metre rule to measure d for one of the discs. She used vernier calipers to measure t for one of the discs.

The student then repeated this process for two more discs and calculated average values for d and t of the discs.

Describe improvements to the student’s method that would give more accurate values for d and t of the discs.

The student determined the average mass of a disc to be 42.5 g ± 0.5 g.

Calculate the density of the material the discs are made from and the percentage uncertainty in this value.

average diameter of disc = 5.2 cm ± 0.1 cm average thickness of disc = 1.9 mm ± 0.1 mm

An object moving through a fluid experiences a drag force due to the viscosity of the fluid.

Some liquid chocolate is poured into a mould. Within the chocolate a bubble of air is formed. The bubble has negligible weight and moves upwards at a constant speed, as shown.

Calculate the constant speed at which the air bubble moves upwards in the liquid chocolate.

radius of air bubble = 6.8 10^–3m density of liquid chocolate = 925 kg m^–3 viscosity of liquid chocolate = 52.6 Pa s

The viscosity of a gas increases with temperature.

It is suggested that the relationship between viscosity η and temperature T is of the form

where A and B are constants.

Explain why a graph of log η against logT would give a straight line.

The table shows data for the viscosity of hydrogen gas over a range

Plot a graph of log η against logT. You should use the columns provided to show any processed data.

Determine values for A and B. You are not required to state the units of A.

A cantilever is a beam fixed at one end.

A student made a cantilever by clamping a metre rule to a bench and attached a mass M at the free end, as shown.

The student displaced the free end through a small distance. When the free end was released, the cantilever oscillated with simple harmonic motion of period T.

The student measured the time for 5 oscillations with an electronic stopwatch. This was repeated twice.

Explain one modification to the apparatus which would improve the accuracy of the data the student obtained.

Explain one modification to the student’s method which would reduce the percentage uncertainty in her measurement of T.

The length of the metre rule overhanging the bench was L. The period of oscillation T is related to L, M and the Young modulus E of the wood of the metre rule by the expression

The student repeated the timing measurements for a half‑metre rule.

The half‑metre rule had the same added mass M, width and thickness as the metre rule.

The metre rule was made from wood of Young modulus E₁, and the half‑metre rule was made from wood of Young modulus E₂. The student collected the following data.

The table below gives the Young modulus values of various types of wood.

Deduce the type of wood each rule is made from.

The potential difference (p.d.) from a power pack alternates at a frequency of 50 Hz. The root-mean-square (r.m.s.) value of the output p.d. is measured to be 8.5 V using an a.c. voltmeter.

The output from the power pack is connected to a resistor of resistance 1.5 Ω.

Show that the rate of power dissipation in the resistor is about 48 W.

A lab technician used the resistor to heat 1.25 10^–4m³ of milk in a beaker.

Calculate the time taken for the milk to increase in temperature by 12 °C.

specific heat capacity of milk = 3930 J kg^–1K^–1

The lab technician used an oscilloscope to determine the supply frequency and check the r.m.s. output p.d. displayed by the voltmeter.

Describe the methods the technician should use.

In an experiment to determine the acceleration of free‑fall, g, a metal cylinder was dropped from rest down a glass tube. A light gate was positioned close to the outside of the glass tube, as shown.

The light gate was connected to a data logger that recorded the time t taken for the cylinder to pass the gate.

A student recorded the following data.

The student omitted t₁ when calculating the mean time. He justified this by stating that t₁ is an anomalous result. Another student stated that the correct mean time should use all the data in the table.

Explain who is correct.

Calculate the percentage uncertainty in the mean time t_mean.

The student used the mean time to calculate the velocity of the metal cylinder as it moved past the light gate.

State the extra measurement the student must have made to calculate this velocity.

The metal cylinder fell through a distance 0.75 m before reaching the light gate.

Calculate the velocity of the metal cylinder as it reached the light gate.

Explain why a light gate and data logger were used in this experiment rather than a stopwatch. You should refer to your answer to (b)‌(i).

A student made measurements to determine if some gold coins were made from pure gold. The coins that were available to the student are shown below.

The student used digital calipers to measure the thickness t and the diameter d of one of the coins.

Calculate the volume V of the coin, and the percentage uncertainty in V.

t = 1.54 mm d = 22.16 mm

The student measured the mass of the coin using an electronic balance. The balance had a resolution of 0.1 g.

Assess whether the coin could be made from pure gold.

density of pure gold = 1.93×10⁴kg m^–3 mass of coin = 11.2 g

The student’s experimental method could have been improved.

Explain two changes the student could have made to the experimental method.

A student used springs to attach a trolley between two fixed supports, as shown.

When displaced horizontally, the trolley oscillated with simple harmonic motion.

To determine the time period T of oscillation of the trolley, the student displaced the trolley from its equilibrium position and released it. As she released the trolley, she started a stopwatch. She stopped the stopwatch when the trolley had returned to its starting point.

Explain how the procedure used by the student to determine T could have been improved.

The mass of the trolley was M. The student added a small mass m to the trolley and determined the new value of T. She repeated the procedure for a range of values of m.

She plotted a graph of T² against m.

Explain how she could use her graph to determine a value for M.

Rosslyn Chapel is a 15th century chapel in Scotland. Inside the chapel, small sandstone cubes protrude from a number of arches. It has been suggested that carvings on these cubes bear a resemblance to standing wave patterns that can be produced on a vibrating metal plate.

A metal plate is made to vibrate and sand is scattered evenly across its surface. At a certain frequency the sand moves to produce the standing wave pattern shown below.

(Source: https://skullsinthestars.com/2013/05/02/physics-demonstrations-chladni-patterns/#jp-carousel-7353)

Explain why the sand moves to different positions when a standing wave is formed on the plate.

The speed of sound in air can be determined by creating a standing wave in a column of air. The diagram shows a tuning fork just above the open end of a tube.

The tuning fork produces a sound wave of known frequency f. Several tuning forks are available, each with a different frequency.

A student adjusted the length L of the air column. A loud sound was heard when a standing wave was produced. A node was formed at the water surface, and an antinode was formed at the open end of the tube.

The student used values of L and f to determine a value for the speed of sound.

Describe a graphical method that the student could have used to determine a value for the speed of sound.

A student carried out an experiment with a light dependent resistor (LDR). He connected the LDR in series with an ammeter and a power supply, as shown in the circuit diagram.

The student placed the LDR a known distance from a desk lamp, as shown.

The student noted the reading on the ammeter as he brought the LDR closer to the lamp.

The student planned to vary the intensity of light incident upon the LDR by adjusting the distance x between the LDR and the lamp.

He thought that the intensity of light on the LDR would increase uniformly if he decreased x by equal intervals. He therefore planned to take ammeter readings as he decreased x from 20.0 cm to 10.0 cm in equal intervals.

Criticise the student’s plan for data collection.

Explain one precaution that the student should take to ensure that results are accurate.

Explain why the ammeter reading increased as the LDR was brought closer to the lamp. Your answer should include reference to the charge carriers in the LDR.

A student connected a voltmeter to a potential divider, as shown in the circuit diagram.

The potential divider was adjusted by moving sliding contact S from position X to position Y.

Explain how the voltmeter reading V depends upon the position of S.

A filament bulb and ammeter were connected to the potential divider. The potential divider was used to vary V across the filament bulb. The ammeter gave the corresponding current I in the filament.

I varied with V, as shown in the graph below.

Explain the shape of the graph.

The temperature T of the filament varies with the potential difference V across the filament according to the expression

where a and b are constants.

Explain why a graph of logT against logV would give a straight line.

Data for T and V is shown in the table below.

Plot a graph of logT against logV on the grid opposite. Use the extra columns provided to show any processed data.

Determine a value for b using your graph.

A student used a laser pointer to direct monochromatic light normal to the plane of a diffraction grating as shown.

A diffraction pattern was produced on the screen. The distance between the first order maximum and the central maximum of the diffraction pattern was x. The distance between the diffraction grating and the screen was D.

The diffraction grating had 300 lines per mm. The laser pen was marked with λ = 520 nm.

Determine whether the spacing of the diffraction pattern was consistent with these values.

x = 43.5 cm D = 2.75 m

The student wrote the following conclusion:

“The value of x was measured with a metre rule. A metre rule has a precision of 0.1 cm, so this value was determined with a high degree of accuracy.”

Comment on this conclusion.

A student investigated the horizontal oscillations of a trolley between two springs, using the apparatus shown.

The student displaced the trolley from its equilibrium position. She then released the trolley and started a stopwatch. She stopped the stopwatch when the trolley had completed one oscillation.

Describe how the method used by the student could be improved to determine a more accurate value of the time period.

The student displaced the trolley 6.0 cm from the equilibrium position. She recorded the velocity of the oscillating trolley using a sensor connected to a data logger.

The output from the data logger is shown below.

The time-base of the data logger output was set to 250 ms div^–1.

Determine the maximum velocity of the trolley.

Explain the shape of the graph.

Determine the effective spring constant k of the oscillating trolley system.

mass of trolley = 0.87 kg

A student investigated the behaviour of small, positively charged oil drops in an electric field.

He introduced an oil drop between two horizontal metal plates. A potential difference V was applied to bring the oil drop to rest as shown.

Add to the diagram to show the polarity of the power supply when the oil drop is at rest.

The oil drop was at rest when V = 4870 V.

The student expected the charge on the oil drop to be a whole number multiple of the charge on an electron.

Deduce whether this is confirmed by the experimental data.

distance between top plate and bottom plate = 1.55 cm

The student determined the terminal velocity of the oil drop in order to obtain a value for the weight of the oil drop.

He disconnected the power supply and the oil drop fell downwards. He viewed the oil drop through a microscope.

Explain why the student should wait a short while before starting to take measurements.

The view through the microscope is shown below. The scale is in millimetres.

The position of the oil drop at the start and end of a 120 s time interval is indicated by the black dots.

Determine the terminal velocity of the oil drop.

The student repeated the measurements on fifty oil drops. For each drop the student calculated the charge on the oil drop.

His results are shown in the bar chart.

The student predicted that the charge on each oil drop would be a whole number multiple of the charge on an electron.

Comment on the extent to which the bar chart supports the student’s prediction.

The student displaced the pendulum through a small angle so that it oscillated. She determined the time period T as 2.50 s.

Calculate the length of the pendulum.

Explain why the amplitude of oscillation of the pendulum did not stay constant.

The student recorded how the amplitude of oscillation varied over time.

It is suggested that the relationship between amplitude A and time t is

where A₀ is the initial amplitude of the oscillation and k is a constant. Explain why a graph of lnA against t would give a straight line.

The table shows the student’s data.

Plot a graph of lnA against t on the grid opposite. Use the additional column to show your processed data.

Determine values for A₀ and k.

The photograph shows some metal washers. A student carried out an experiment to determine the density of the metal the washers are made from.

Each washer has a diameter d₁ of about 4.5 cm. The internal diameter d₂ of each washer is about 2.5 cm. Each washer has a thickness t of about 4 mm.

The student used a half metre rule to make measurements of a washer.

Comment on the student’s choice of measuring instrument.

The student measured t for each of the five washers and then calculated a mean value.

Explain how the student could modify her method to obtain a more accurate mean value for t.

Show that the percentage uncertainty in her value for the area of a washer is about 3%.

The student obtained the following values of t for each of the five washers.

The table shows the density of iron and steel.

Deduce whether the washers are made from iron or steel.

mean mass of a washer = 32.0 g

The uncertainty in the mass is negligible.

A student investigated the behaviour of a spring under tension. The spring was hung vertically with a mass holder attached, as shown. The position of the bottom of the mass holder was determined using a metre rule.

The position of the bottom of the mass holder was recorded. The spring was stretched by adding masses to the mass holder and the new positions were determined. The extension of the spring for each mass was calculated.

The results are shown in the table.

Criticise the recording of these results.

Describe how the student should determine the extension of the spring as accurately as possible.

A student was given a box of identical glass microscope slides and asked to determine the density of the glass. She used a micrometer to measure the thickness of one of the slides. She repeated this measurement twice in different places and calculated a mean value for the thickness. The thickness of each slide was approximately 1 mm.

Explain how she should have measured the thickness of the slides in order to minimise the percentage uncertainty.

In her report she wrote

“My value for the mass of the glass slides was precise, because I measured the mass using an electronic balance which was accurate to the nearest 0.01 g. I reduced the effect of random error by repeating the measurement several times.”

Comment on this statement.

The diagram shows a metronome, which includes an inverted pendulum, used by musicians to set a tempo. The pendulum oscillates with simple harmonic motion and makes a loud click at regular intervals.

A faulty metronome stopped making a clicking noise. A student tried to check the accuracy of the period T of the metronome, using a stopwatch. The student timed the pendulum as it moved from one extreme of the oscillation to the other.

Explain how the procedure used by the student to determine T could have been improved.

A student carried out an experiment to calibrate a thermistor. She connected the thermistor in series with a resistor and a power supply as shown. Then she placed the thermistor in a beaker of hot water and used a thermometer to record the temperature θ of the water.

The student recorded θ and corresponding values of the reading V on the voltmeter as the water cooled.

Explain, making reference to charge carriers, why V increased as the water cooled.

Over a limited temperature range V varies with θ according to the expression

where b and V₀ are constants.

Explain why a graph of lnV against θ would give a straight line.

The student’s data is shown in the table below.

Plot a graph of lnV against θ on the grid opposite. Use the column provided to show any processed data.

Determine values for b and V₀.

A student used a sonometer to investigate the properties of a stretched wire. The sonometer is a long hollow wooden box.

A steel wire is attached to one end of the box and rests on two wooden bridges. The wire is placed under tension T by hanging a mass from the end of the wire, as shown.

The student placed the base of a vibrating tuning fork in contact with the wire, at one of the bridges. This set the wire into oscillation. He adjusted the position of the other bridge until a single-loop standing wave was produced on the wire between the bridges.

Explain how an antinode is produced at the mid-point of the wire between the bridges.

The student repeated this for a series of tuning forks with different frequencies f. For each fork he measured the distance L between the bridges.

The steel wire, of mass per unit length μ, was placed under tension T by hanging a mass of 2.10 kg from the end of the wire.

State one safety precaution that should be taken when carrying out the investigation.

The student plotted a graph of L² against 1/f². Show that the gradient of this graph is equal to

The student then found a value of μ for a brass wire, using a different method.

He measured the diameter d of the wire using a micrometer.

Explain one technique the student should use when measuring d.

The student obtained the following data.

The stated value of μ for the brass wire used by the student was

Deduce whether the student’s data supports this value for μ.

density of brass = 8700 kg m⁻³ ± 200 kg m⁻³

The Planck constant can be determined in a school laboratory using light emitting diodes (LEDs).

An LED emits light when the potential difference (p.d.) across it is large enough to transfer sufficient energy to an electron to result in the emission of a photon. The electron must have energy greater than or equal to the photon energy.

The minimum p.d. required to produce light from LEDs emitting different frequencies was measured by increasing the p.d. from zero until light was first seen.

The graph shows the results.

Determine the value of the Planck constant given by this graph.

There are two problems with using LEDs to determine the Planck constant:

• when the p.d. is increased and the LED first emits light it is difficult to see

• the LEDs do not emit a single frequency but also light of frequencies slightly above and below the recorded frequency.

Discuss the extent to which these problems are consistent with obtaining a result from this graph for the Planck constant which is higher than the accepted value.

The length l of wire in the magnetic field was 15.5 cm. When the current in the circuit was 4.55 A, the reading on the balance increased by 5.65 g.

Calculate the magnetic flux density B in the region of the rigid wire.

The student wrote the following statement

“The balance could read to the nearest 0.01 g, which makes my values for the magnetic force both accurate and precise.”

Comment on this statement.

A student used a Geiger-Müller (GM) tube to determine the activity of a radium source. Radium emits α, β, and γ radiation.

He positioned the source 20 cm from the GM tube, as shown, and recorded the count for 1 minute. He repeated the measurement and calculated a mean count.

The student recorded the following results.

Criticise the student’s method for determining the count at this position.

From his results the student determined that the activity of the source was 3.0 Bq.

Comment on his value for the activity of the source.

Genuine crystal balls are made from clarified quartz rather than glass. A student was given a small crystal ball and wanted to know whether it was genuine.

The mean diameter of the crystal ball was measured to be 5.06 cm and the mass of the crystal ball was measured to be 175 g.

Show that the density of the material of the crystal ball is about 2600 kg m⁻³.

The student measured the diameter of the crystal ball using vernier calipers with a resolution of 0.01 cm. She measured the mass of the crystal ball using a balance with a resolution of 1 g.

The table gives the densities of clarified quartz and glass.

Determine whether the crystal ball was genuine.

A student measured the height h of a liquid column in a capillary tube. She used a travelling microscope to make measurements of the positions of the top and bottom of the liquid column.

The travelling microscope consists of a simple microscope that can be moved vertically along a vernier scale.

The student used a capillary tube with an internal radius r equal to 0.10 mm and recorded the following readings from the vernier scale.

State the uncertainty in each of these readings.

Calculate the percentage uncertainty in the student’s value of h.

The student repeated the measurement of h for capillary tubes of different radii.

The table shows the student’s final data.

Criticise the student’s recording of the data.

The student plotted the following graph.

Determine the height of the liquid column that the student could expect for a tube with an internal radius of 0.11 mm.

In her notes it stated that

Assess the extent to which the student’s data supports this relationship.

A student connected the output from a source of alternating potential difference (p.d.) to a series resistor combination.

She connected an oscilloscope across the 150 Ω resistor as shown.

The trace obtained on the oscilloscope is shown below.

Determine the peak p.d. across the 150 Ω resistor.

y-sensitivity of oscilloscope = 2.0 V per division

Peak p.d. across 150 Ω resistor =_........

Calculate the root mean square (r.m.s.) value of the current in the circuit.

r.m.s. value of current =_........

Calculate the power dissipated in the circuit.

Another student suggested that a voltmeter would be more accurate than using an oscilloscope to determine the magnitude of the p.d.

Comment on this suggestion.

A Hertzsprung-Russell (HR) diagram shows how the luminosity L depends on the surface temperature T for a group of stars.

The HR diagram below is for a young star cluster.

Explain how we can tell that the young star cluster is in the early stages of its evolution.

Explain why the most massive stars in the cluster have the greatest luminosities.

The HR diagram on the previous page shows an approximately linear relationship for stars in this cluster.

It is suggested that the relationship between luminosity L and surface temperature T is of the form

where k and n are constants.

Explain why a graph of log L against log T would give a straight line.

The table shows data for stars in this cluster.

Plot a graph of log L against log T on the grid opposite. Use the columns provided to show any processed data.

Determine a value for n.

A student investigated the behaviour of a spring under tension. The spring was hung vertically with a mass holder attached as shown.

The student measured the length of the spring as he added masses to the holder. The rule was held as shown to measure the distance between the top and bottom coils of the spring. He determined the extension for each value of total mass on the holder. He did this by subtracting the original length of the spring from each extended length.

Explain whether this method would produce accurate values for the extensions of the spring.

Explain how the student could modify his method in order to obtain more accurate values for the extensions of the spring.

In another experiment, the student displaced the mass vertically each time a mass was added to the spring. He used a stopwatch to determine the period of vertical oscillations of each mass.

The student used his data to plot a graph of T² against m as shown.

The student expected the graph to be a straight line through the origin. He thought that there may be systematic error due to reaction time.

Give an example of another possible systematic error in this experiment.

Another student suggests that to reduce the uncertainty in the value for the period, a data logger connected to a light gate could be used to measure time.

Comment on the student’s suggestion.

Determine a value for the stiffness of the spring.

When determining the period of oscillation for each mass, the student measured the time for 20 oscillations. He repeated this measurement to obtain a mean time for 20 oscillations.

Explain how the student’s procedure contributed to the accuracy of the measurement.

Light from a laser pointer was passed through a diffraction grating. The light was perpendicular to the diffraction grating as shown. A diffraction pattern was produced on

The distance between the first order maximum and the central maximum of the diffraction pattern was x. The distance between the diffraction grating and the screen was D.

Distance x was measured to be 0.500 m with a metre rule. The wavelength of light λ₁ from the laser pointer was 650 nm. The laser pointer was replaced with one that produced light of a different wavelength. The new distance x was measured to be 0.400 m.

D = 1.45 m

Calculate the wavelength λ₂ of the light emitted by the replacement laser pointer.

Explain one modification to this method that would decrease the uncertainty in the

In another experiment, the light from the laser pointer was not quite perpendicular to the screen.

Explain how this would change the diffraction pattern produced on the screen.

A teacher is explaining the differences between accuracy and precision to her students. She draws the following diagram, which shows different degrees of accuracy and precision. The circles represent targets A, B, C and D and the dots represent arrows hitting the targets.

Explain how targets A, B, C and D represent differing degrees of accuracy and precision.

A student released a ping pong ball in front of a metre rule and used a phone camera to record the motion of the ball as it fell. The phone camera captures 60 images per second, which may be played back one image at a time.

The ball was dropped from a height such that it reached its terminal velocity as it passed the metre rule.

Explain how the terminal velocity of the ball could be determined using the phone camera recording.

Explain how a systematic error could affect the value obtained for the terminal velocity.

This experiment could have been attempted using a stopwatch to measure the time as the ping pong ball fell.

Explain an advantage of using a phone camera rather than a stopwatch.

A student released a ping pong ball in front of a metre rule and used a phone camera to record the motion of the ball as it fell. The phone camera captures 60 images per second, which may be played back one image at a time.

The ball was dropped from a height such that it reached its terminal velocity as it passed the metre rule.

Explain how the terminal velocity of the ball could be determined using the phone camera recording.

Explain how a systematic error could affect the value obtained for the terminal velocity.

This experiment could have been attempted using a stopwatch to measure the time as the ping pong ball fell.

Explain an advantage of using a phone camera rather than a stopwatch.

A student carried out an experiment with a pendulum hung from a fixed support. The fixed support was a distance h above floor level as shown.

As the student was unable to measure the length of the pendulum directly, she measured the distance d from the bob to the floor.

To determine the period T of the pendulum, the student used the following method:

• release the bob from its highest position and start a stopwatch • stop the stopwatch when the bob reaches the same position again.

Criticise the student’s method for measuring the period.

The student used her data to plot a graph of T² against d as shown below.

Determine a value for the acceleration due to gravity g.

A student carried out an experiment with a pendulum hung from a fixed support. The fixed support was a distance h above floor level as shown.

As the student was unable to measure the length of the pendulum directly, she measured the distance d from the bob to the floor.

To determine the period T of the pendulum, the student used the following method:

• release the bob from its highest position and start a stopwatch • stop the stopwatch when the bob reaches the same position again.

Criticise the student’s method for measuring the period.

The student used her data to plot a graph of T² against d as shown below.

Determine a value for the acceleration due to gravity g.

A student investigated the rate at which a hot liquid transfers thermal energy to the surroundings. He placed hot water in a Pyrex beaker and measured the temperature of the water using a liquid-in-glass thermometer.

He obtained the following data for the temperature θ of the water at times t. He measured

temperature of surroundings = 23 °C

Theory suggests that a liquid transfers internal energy to the surroundings at a rate proportional to the temperature difference Δ θ between the liquid and the surroundings.

This leads to the expression

where b is a constant and Δθ₀ is the initial temperature difference.

Explain why a graph of ln Δθ against t should be a straight line.

Plot a graph of ln Δθ against t on the grid opposite. Use the columns provided in the table to show any processed data.

Determine the value of b.

The student suggested that the experiment would have been more accurate if a temperature sensor and data logger had been used to collect the data.

Assess the validity of the student’s suggestion.

A student investigated the rate at which a hot liquid transfers thermal energy to the surroundings. He placed hot water in a Pyrex beaker and measured the temperature of the water using a liquid-in-glass thermometer.

He obtained the following data for the temperature θ of the water at times t. He measured

temperature of surroundings = 23 °C

Theory suggests that a liquid transfers internal energy to the surroundings at a rate proportional to the temperature difference Δ θ between the liquid and the surroundings.

This leads to the expression

where b is a constant and Δθ₀ is the initial temperature difference.

Explain why a graph of ln Δθ against t should be a straight line.

Plot a graph of ln Δθ against t on the grid opposite. Use the columns provided in the table to show any processed data.

Determine the value of b.

The student suggested that the experiment would have been more accurate if a temperature sensor and data logger had been used to collect the data.

Assess the validity of the student’s suggestion.

A simple loudspeaker consists of a cone, a coil of wire and a magnet. The cone and coil are attached to each other and are free to move. An alternating current in the coil causes the cone to oscillate. The loudspeaker is mounted in a wooden box. A cross-section through the loudspeaker is shown.

A student made the following observations:

• when an alternating potential difference (p.d.) is applied to the coil, the cone oscillates

• the frequency of oscillation is the same as the frequency of the p.d.

• at particular frequencies, the box vibrates with a large amplitude.

*(a) Explain these observations.

Explain why the sand collects in heaps.

The student determined the distance d between the centres of adjacent heaps.

Describe the procedure she should follow to determine an accurate value for d.

Assess whether the experimental data is consistent with a value for the speed of

signal generator frequency = 3.25 kHz.

d = 5.1 cm

A simple loudspeaker consists of a cone, a coil of wire and a magnet. The cone and coil are attached to each other and are free to move. An alternating current in the coil causes the cone to oscillate. The loudspeaker is mounted in a wooden box. A cross-section through the loudspeaker is shown.

A student made the following observations:

• when an alternating potential difference (p.d.) is applied to the coil, the cone oscillates

• the frequency of oscillation is the same as the frequency of the p.d.

• at particular frequencies, the box vibrates with a large amplitude.

*(a) Explain these observations.

Explain why the sand collects in heaps.

The student determined the distance d between the centres of adjacent heaps.

Describe the procedure she should follow to determine an accurate value for d.

Assess whether the experimental data is consistent with a value for the speed of

signal generator frequency = 3.25 kHz.

d = 5.1 cm

A spring is made from loops of thick steel wire as shown.

There are two extra loops, one on each end of the spring.

A student determined the length of steel used to make the spring by using vernier calipers to measure the width w of the spring. The length of wire l on each loop is given by l = πw

The student obtained the following values for w.

Calculate l.

Estimate the percentage uncertainty in your value for l.

Calculate the total length L of wire used to make the spring.

The student measured the diameter d of the steel wire and obtained a value of 2.52 mm.

Explain which instrument he used to measure the diameter.

Estimate the percentage uncertainty in the student’s value for d.

The student used a balance to measure the mass m of the spring. He obtained a value of 32.0 ± 0.5 g.

Estimate the percentage uncertainty in the mass of the spring.

Determine whether the data collected leads to a value for the density of steel in agreement with the standard value.

density of steel = 7 800 kg m⁻³

A spring is made from loops of thick steel wire as shown.

There are two extra loops, one on each end of the spring.

A student determined the length of steel used to make the spring by using vernier calipers to measure the width w of the spring. The length of wire l on each loop is given by l = πw

The student obtained the following values for w.

Calculate l.

Estimate the percentage uncertainty in your value for l.

Calculate the total length L of wire used to make the spring.

The student measured the diameter d of the steel wire and obtained a value of 2.52 mm.

Explain which instrument he used to measure the diameter.

Estimate the percentage uncertainty in the student’s value for d.

The student used a balance to measure the mass m of the spring. He obtained a value of 32.0 ± 0.5 g.

Estimate the percentage uncertainty in the mass of the spring.

Determine whether the data collected leads to a value for the density of steel in agreement with the standard value.

density of steel = 7 800 kg m⁻³

An engineer was checking the dimensions of a steel tube. The tube had a length of about 45 mm and an external diameter of about 15 mm as shown.

She used a digital micrometer to measure the diameter of the tube. Before taking the reading she closed the jaws of the micrometer to check for a zero error.

State the type of error she avoided by doing this.

Describe the procedure she should follow to determine an accurate value for the external diameter of the tube.

The engineer determined the length of the tube using the micrometer. The reading on the micrometer scale was 45.043 mm. She recorded the reading as 45.0 mm. State why recording a reading of 45.043 mm could not be justified.

A student determined the specific heat capacity of aluminium.

She used an electrical heater to heat an aluminium block and measured the temperature of the block with a digital thermometer.

She connected the electrical heater into a circuit and took measurements to determine the power of the heater.

Draw a circuit diagram of a suitable circuit.

The student monitored the temperature θ of the aluminium block over the time t for which the heater was switched on.

Her results are plotted on the graph.

Determine the specific heat capacity of aluminium.

power of heater = 37.5 W mass of aluminium block = 0.986 kg

The student looked up the accepted value for the specific heat capacity of aluminium. Using this value, the student predicted that it should have taken 240 s for the temperature of the aluminium block to increase by 10 °C.

Explain the difference between the predicted time and the student’s actual observations.

A student was investigating the forces involved in circular motion.

He placed a small coin on a horizontal turntable as shown. The turntable was connected to a driver unit so that it could be rotated at a constant rate.

The student switched on the driver unit and increased the rate of rotation until the coin slid off the turntable. He read the angular velocity of the turntable from a digital display on the driver unit. He then replaced the coin in the original position on the turntable and repeated the procedure.

His results are shown.

The student used the results to calculate a mean value of .

State the purpose of calculating a mean.

Calculate the percentage uncertainty in the mean value of .

The student used and r to calculate the centripetal acceleration of the coin at the instant it started to slide.

Calculate the percentage uncertainty in this centripetal acceleration.

The student repeated the procedure with different values of r.

Explain how the value of at which the coin started to slide varied as r increased.

A trolley is attached to the ends of two springs as shown. When displaced from its equilibrium position, the trolley moves with simple harmonic motion.

A student has a stopwatch and metre rule available.

Explain the procedure that the student should follow to make an accurate determination of the time period T of the trolley.

Describe how the student should use her value of T to determine the maximum speed of the trolley.

Another student suggests that a more accurate value for T could be obtained by using a position sensor and data logger.

Comment on this suggestion.

The student displaces the trolley a greater distance from the equilibrium position, so the amplitude of oscillation is doubled. The trolley still moves with simple harmonic motion.

Explain how the maximum kinetic energy of the trolley will change.

The Beaufort scale is used to describe wind intensity. On this scale the average wind speed v increases with the Beaufort scale value B.

The relationship between v and B is given by

where k and p are constants.

Explain why a graph of log v against log B should give a straight line.

The table gives some values of v and corresponding values of B.

Plot a graph of log v against log B on the grid opposite. Use the columns provided to show any processed data.

Determine the values of p and k.

A student investigated how the volume of a fixed mass of air varies with the temperature of the air. She used the apparatus shown.

A glass tube was sealed at one end. A plug of oil trapped a length l of air in the tube. The water in the beaker was heated to a temperature . The corresponding value of l was measured. This was repeated for a range of temperatures.

The thermometer had a resolution of 0.5 °C. The scale had mm divisions.

The student’s results are shown in the table.

/ °C	l / cm
24	8.8
60	9.8
78.5	10.3
95.5	10.9

Criticise the student’s results.

Explain two possible sources of error in this investigation.

Describe two improvements that would increase the accuracy of measurements obtained in this investigation.

The student plotted a graph of l against θ as shown.

Explain the significance of the intercept on the x-axis.

The student wrote a report of the investigation in her lab book. In the conclusion she wrote:

“In this investigation uncertainties were minimised by selecting measuring instruments with a high resolution. The points lie on a perfect straight line, indicating that the investigation is accurate.”

Discuss the student’s conclusion.

Impact craters are formed when meteorites strike the surface of a planet. A student investigated some factors that might influence the formation of impact craters. He did this by dropping spheres of modelling clay into a tray of sand.

The diameter of the crater produced by each sphere was measured using vernier calipers as shown.

This process was repeated for spheres of different diameters.

In one test, the spheres were dropped from the same height.

Determine the factor by which the kinetic energy of the sphere just before impact increases when the sphere diameter is increased from 2.0 cm to 4.0 cm.

A practical physics textbook states that “measurements may give a precise value for the quantity being determined but this may not necessarily be an accurate value”.

Describe what physicists mean by the terms accuracy and precision.

The temperature of the air in a room is measured using a mercury-in-glass thermometer.

Describe how the value for the temperature may be precise but not accurate.

A student is investigating how the internal resistance of a dry cell varies over time. She sets up the circuit shown to draw current from the cell.

The student proposes to use a data logger to monitor the terminal potential difference (p.d.) of the cell over a period of time.

State why a data logger would be suitable to collect data in this investigation.

The graph shows how the terminal p.d. varies as the chemical energy store in the cell is depleted.

Explain, using the graph, why the current drawn from the cell decreases as the chemical energy store in the cell is depleted.

Determine the internal resistance of the cell when its chemical energy store is 80% depleted. Assume that the e.m.f. of the cell remains constant at 4.0 V.

State a suitable measuring instrument that could be used with a single disc to measure t.

A balance which can measure mass with a resolution of 0.2 g is available.

Determine the minimum number of discs that should be placed on the balance together if the percentage uncertainty in the measurement of the mass is to be less than 0.5%.

density of steel = 7900 kg m^–3

The measured uncertainty in d is ±0.1 mm and the measured uncertainty for t is ±0.05 mm.

Determine the percentage uncertainty in the calculated volume of the disc.

A student set up the circuit shown and measured the current I through the filament lamp for a range of values of potential difference (p.d.) V.

The student’s data is shown in the table.

Criticise the student’s recording of the data.

The student drew a graph of how current varies with p.d. She drew a straight line on the graph and claimed that the data demonstrates that the filament lamp obeys Ohm’s law because the graph is linear.

Assess the validity of the student’s statement.

Using the circuit shown the student was unable to obtain data for p.d.s less than 2.5 V.

Draw a diagram of a circuit the student could have used to enable a full range of p.d.s from 0 to 12 V to be investigated.

A student is using a simple pendulum to determine a value for the acceleration of free fall g.

She measures the length l of the pendulum four times with a metre rule and records the following values.

She calculates the mean length l_m of the pendulum using the following method:

Calculate a more accurate value for l_m.

Determine the time period of the oscillations of this pendulum, using your^m calculated value for l_m.

She sets the pendulum into oscillations with small amplitude and uses a stopwatch to determine the time period.

The student releases the pendulum at A and simultaneously starts the stopwatch. She measures the time taken for 5 oscillations and divides the value by 5. She repeats the procedure twice and calculates a mean time period.

Explain two modifications to the student’s method that would improve the value obtained for the time period.

The International Space Station (ISS) is in a low Earth orbit. Astronauts in ISS have an apparent weight of zero. In order to determine their mass, the astronauts must secure themselves to a platform which is set into oscillation and moves with simple harmonic

Explain why the astronauts in the ISS have an apparent weight of zero.

State what is meant by simple harmonic motion.

Describe how, using a stopwatch and a ruler, the following quantities could be determined for the oscillating platform:

the frequency of oscillation

the maximum speed of the platform.

The platform continues to move with simple harmonic motion at the same frequency, but its amplitude is doubled.

Explain how the maximum kinetic energy of the platform will change.

A method to determine the wavelength of light using a converging lens was first proposed by Sir Isaac Newton.

A converging lens is placed on a plane glass plate. The lens is illuminated from above with a parallel beam of monochromatic light, as shown.

Some of the light is reflected from the upper surface of the lower glass plate and some from the lower surface of the lens. Interference between these two reflected waves produces circular fringes. The pattern is viewed through a microscope.

The diameter D of each circular fringe, numbered N from the centre, is measured using the microscope. The data obtained from such an experiment is shown.

The relationship between N and D is of the form D = pN^q where p and q are constants.

Determine p and q for this data using a graphical method. Use the additional columns for your processed data.

The table below shows the readings from which the diameter of the first dark circle was calculated.

Use these readings to estimate the percentage uncertainty in the diameter due to the resolution of the instrument.

State why the actual percentage uncertainty would have been greater than the value calculated in (b)(i).

When considering the principles of this experiment, a student suggests that interference fringes would only be produced with monochromatic light. This is because interference requires coherent light waves.

Discuss the validity of the student’s suggestion.

A student investigates the motion of a friction‑free trolley down a ramp. On the top of the trolley there is a metal loop which makes contact with a length of thin resistance wire, AB, fixed above the ramp. The resistance wire has a uniform diameter.

The trolley accelerates down the ramp and the metal loop stays in contact with the wire along the full length of the ramp.

The student uses a protractor to measure the angle θ between the ramp and the horizontal

and records a value of 4° with an uncertainty of ±1°.

The two ends of the wire are connected to a 1.5 V cell. A data logger, set to measure potential difference, is connected to the metal loop and to the negative terminal of the cell.

Explain how the potential difference recorded by the data logger will vary as the loop moves along the length of the wire AB.

The graph shows the data obtained from the data logger.

Determine the velocity of the trolley at 1.5 s.

1.5 V represents a distance of 2.00 m.

The student calculated the velocity of the trolley at 2.0 s to be 1.5 m s^–1.

By considering the acceleration of the trolley, determine whether the student’s measurement of θ was within the uncertainty quoted.

A physics textbook states that “when carrying out experimental measurements there will always be errors and uncertainties”.

Describe what physicists mean by error and by uncertainty.

Give two reasons why a measurement may have an uncertainty.

A student carries out measurements to determine the density of glass. The student has 20 glass microscope slides available.

The approximate dimensions of one slide are shown.

The density is calculated using the equation

Describe how the student can determine an accurate value for the density of the glass. Your answer should include the measuring instruments required.

State one precaution that the student should take to ensure the measurements are accurate.

A student set up a “seconds pendulum”. This is a simple pendulum for which the time taken to move from the bob’s highest position on one side to its highest position on the opposite side is 1.00 s.

Calculate the length L required for the pendulum to be a “seconds pendulum”.

The student set the pendulum into oscillation. She used a stopwatch to check the accuracy of the pendulum’s period T.

Describe the procedure the student should have used to obtain an accurate value for T.

Another student suggested that the uncertainty in the measurement of the time period of the pendulum could be reduced by using a light gate and a data logger. The data logger would record the time between successive interruptions of the light beam. Both the data logger and the stopwatch have a resolution of 0.01 s.

Comment on the student’s suggestion of using a data logger rather than a stopwatch.

A student carried out an experiment to determine the resistivity of nichrome wire.

He used an ohmmeter to measure the resistance of a length of nichrome wire as shown.

The diameter of the wire was measured as 0.27 mm ± 0.01 mm. The length of the wire was measured as 1.25 m ± 0.05 m.

Determine which of the three measurements introduces the greatest uncertainty into the value for the resistivity. Your answer should include calculations.

Explain how the student could reduce the uncertainty in the measurement of the diameter.

Calculate the minimum value of resistivity possible from the student’s data.

A teacher is demonstrating how to measure the speed of sound in a steel rod. The equipment comprises a hanging steel rod and a hammer connected to a data logger as shown.

The rod is tapped at A with the hammer. A compression pulse travels to B and is reflected back. When the reflection reaches A the hammer loses contact with the rod. Whilst the hammer is in contact with the rod the output from a 5 V supply is recorded by the data logger.

The graph shows the output from the data logger for one hammer tap.

Explain why a data logger is appropriate for this demonstration.

Use the graph to show that the speed of the pulse in the rod is about 6000 m s⁻¹.

length of steel rod = 1.18 m

The speed of sound v in the rod depends on the Young modulus E and the density ρ of the material of the rod as given by the equation

Calculate the Young modulus of steel.

ρ_steel= 7850 kg m⁻³

The majority of stars in the universe are thought to be main sequence stars. For such stars the luminosity increases with the mass of the star.

It is suggested that the relationship between luminosity and mass is of the form

where L = luminosity, M = (mass of star / mass of the Sun) and L_Sun and p are constants. Explain why a graph of log L against log M would give a straight line.

The table shows data for a range of main sequence stars.

Plot a graph of log L against log M. You may use the columns provided to show any processed data.

Determine values for p and L_Sun and hence state the mathematical relationship between L and M.

A student investigated the way in which gamma radiation spreads out from a source. He placed a cobalt-60 source in a source holder and set up a Geiger-Müller (GM) tube a short distance d away. He connected the GM tube to a counter as shown.

The student recorded the count for 2 minutes.

Describe how to determine the corrected count rate from the source.

His teacher turned the GM tube through 90° so that the side of the tube faced the source as shown below.

Explain why this arrangement could lead to more accurate data.

Explain another modification to the experimental method which would improve the accuracy of the data.

The variation in the intensity of gamma radiation with distance from a point source should obey an inverse square law. If this is the case, then the count rate C should vary with d according to the equation

where K is a constant. The student plotted against d and obtained the following graph.

Draw a line of best fit on the graph.

The student concluded that the graph was consistent with the gamma radiation intensity obeying an inverse square law.

Discuss the extent to which the data obtained supports the student’s conclusion.

It is suggested that the investigation into the way in which gamma radiation spreads out from a source, using the apparatus as shown in (b), could be carried out successfully using a radium-226 source.

Radium-226 emits α, β and γ radiation.

Justify this suggestion.

A student investigated the behaviour of a spring under tension. The spring was hung vertically with a mass holder attached.

The position of the bottom of the mass holder was recorded. The spring was stretched by adding masses to the mass holder and the new positions were recorded. The extension of the spring each time was calculated.

The student produced the following table.

Criticise the student’s table.

The student used her data to plot a graph as shown.

Determine a value for the force constant k of the spring.

Two identical springs are joined in series and a bar magnet is hung from one end as shown.

The bar magnet is displaced a small distance vertically from its equilibrium position and released.

Calculate the frequency at which the system oscillates.

mass of magnet = 120 g spring constant of each spring = 22 N m⁻¹

*(d) Identical bar magnets are suspended from identical springs, with the North pole of each magnet inside a coil of wire as shown. The two coils are connected together with conducting leads.

Magnet A is displaced so that it oscillates vertically. The North pole of magnet A moves into and out of the coil of wire with simple harmonic motion. As this motion continues, magnet B starts to oscillate. The amplitude of oscillation of magnet B increases over time.

Explain why magnet B starts to oscillate with an increasing amplitude.