Errors and uncertainties

Figure 3 shows a circuit that causes a loudspeaker to emit ‘clicks’ at a constant rate. The time T between successive clicks depends on the resistances R₁ and R₂ and the capacitance C. You do not need to know any details about the electronic system.

A student uses a stopwatch to determine T. Table 1 shows the student’s repeated measurements of 20 T.

Suggest why the student measures 20 T.

Suggest why the student repeats their measurements of 20 T.

Determine the percentage uncertainty in the student’s result for T. Assume that all the data in Table 1 are valid.

The student determines T for different values of R₂. The values of R₁ and C are not changed.

The student produces the graph in Figure 4.

It can be shown that

The student has access to a range of fixed and variable resistors. The student wants to modify the circuit in Figure 3 so that T can be varied continuously from a minimum of 0.70 s to a maximum of 1.25 s.

Show how the student can achieve this by combining two resistors to make R₂. You should use information from Figure 4.

This question is about the vertical oscillations of a test tube in a liquid.

Figure 10 shows some wire wound tightly around a test tube. The ends of the wire are secured with tape.

P and Q are points marked on the wire, as shown in the enlarged view in Figure 10. A travelling microscope with a vernier scale is used to measure the distance between P and Q. The microscope reading at P is 9.26 mm.

Figure 11 shows the microscope reading at Q.

Show that the wire has a thickness of approximately 1.2 mm.

The external diameter of the test tube is 18.0 mm.

Deduce the length of wire between P and Q.

A student investigates vertical oscillations of the test tube in a liquid.

Figure 12 shows the test tube and wire floating at rest in a beaker of dark-coloured oil. The part of the test tube below the surface cannot be seen clearly through the oil.

The test tube is then pushed downwards as far as possible, as shown in Figure 13.

The test tube is now released from the position shown in Figure 13. The test tube oscillates vertically with simple harmonic motion of period T.

A horizontal fiducial mark is used to reduce uncertainty in the measurement of T. The fiducial mark is to be placed level with one of the positions A, B, C or D shown in Figures 12 and 13.

Identify the correct position for the fiducial mark. Go on to explain why you chose this position.

When the total mass of the test tube, wire and tape is constant, it can be shown that:

When the test tube oscillates in the oil, T is 0.587 s.

The test tube is cleaned and placed in an identical beaker containing milk. When the test tube oscillates in the milk, T is 0.534 s.

Determine the density of the milk.

Due to damping, only a small number of oscillations occurs after the test tube is released.

The student wants to produce a greater number of oscillations for a particular liquid without changing T.

State and explain two ways in which the apparatus can be modified to achieve this.

This question is based on a method to determine the resistivity of a wire (required practical activity 5).

Figure 1 shows a micrometer screw gauge.

Figure 2 shows an enlarged view of the scales.

State, in mm, the resolution of the main scale.

What is the reading on the micrometer?

Tick () one box.

A wire X is placed in the gap between the anvil and the spindle.

State and explain how this gap is closed just before taking a reading of the diameter of X.

Figure 3 shows a circuit used to determine the resistance per metre of wire X.

Two terminals are used to mount X on a ruler. Clips are used to connect a voltmeter across the 1.2 Ω resistor. When the switch is closed, the voltmeter reading is 931 mV.

The switch is then opened and the voltmeter is connected to X as shown in Figure 4.

When the switch is closed, the voltmeter reading is 397 mV.

Show that, for the arrangement in Figure 4, the resistance R of the wire between the clips is about 0.5 Ω.

The length of wire between the clips is L. Values of R are determined for different values of L. Figure 5 shows these data.

Determine the resistance per metre of X.

Table 1 shows the resistance per metre of various metal wires. The diameter of X is one of the values of d shown in Table 1.

Identify the metal used for X. Go on to determine the resistivity of the metal. State an appropriate SI unit for your answer.

A student adds error bars for R and L to each point on Figure 5.

She estimates that

• each value of R has a percentage uncertainty of 6%

• each value of L has an absolute uncertainty of 5 mm.

Compare her error bars for the point at L = 209 mm with her error bars for the point at L = 388 mm.

Outline how error bars are used to determine the uncertainty in the gradient of a linear graph.

Figure 6 shows apparatus used to investigate how the resistance R of a light-dependent resistor (LDR) varies with illumination.

The ohm-meter

• always displays a four-digit reading of R

• can be set to the different ranges A to E shown in Table 2.

Explain why the reading displayed in Figure 6 shows that the ohm-meter is set to range C.

The quantity E_V is a measure of the intensity of the light incident on the LDR.

The resistance R of the LDR is given by

Show that E_V for the arrangement shown in Figure 6 is about 130 lx.

R is recorded for different values of the vertical distance x between the lamp and the LDR. E_V is calculated for each value of R.

Figure 7 shows how E_V varies with x.

It can be shown that E_{V 2}

Describe a method to show that Figure 7 confirms this relationship. You do not need to show any calculations.

Deduce the value of x when E_V = 130 lx.

R is measured when x = 450 mm. Figure 8 shows how the ohm-meter displays the values of R when set to range B and when set to range C.

The uncertainty of the reading on the ohm-meter is ±2% of the displayed reading plus ±2 in the least significant digit.

This means that:

• using range B the maximum value of R is 1.02 × 1681 + 2 = 1717 Ω

• using range C the minimum value of R is 0.98 × 1.68 – 0.02 = 1.63 kΩ.

Complete Table 3. Go on to explain whether range B or range C should be used to measure R.

Figure 9 shows the LDR being used to investigate the transmission of light through glass slides.

The lamp and ohm-meter are switched on. R is recorded with different numbers of slides placed on the LDR. E_V is calculated for each value of R.

The positions of the lamp and the LDR are not changed during the experiment.

Identify two other control variables.

For the arrangement in Figure 9 it can be shown that

where N is the number of slides μ is a constant.

Explain how μ can be determined from a linear graph.

In an experiment μ = 9.0 × 10⁻²

Deduce the minimum number of slides needed to reduce E_V by 50%.

Figure 7a shows the front view of a vertical coil mounted on a circular frame.

Figure 7b is a side view showing a section through the frame and coil. A constant direct current in the coil produces magnetic flux represented by the magnetic field lines on this diagram.

Figure 7a Figure 7b

Point Q is at the centre of the coil. A sensor placed at Q detects B_H, the horizontal component of the magnetic flux density. The effect of the Earth’s magnetic field at Q is negligible.

Discuss whether a search coil is a suitable sensor to detect B_H.

B_H is measured at Q with the coil vertical.

The coil is now rotated about Q through 25° as shown in Figure 8. The current in the coil does not change.

A new measurement of B_H is made with the coil fixed in this new position.

Determine the percentage change in B_H produced by this rotation of the coil. Show your working.

Figure 9 shows a protractor being used to measure the angle through which the coil is rotated.

Estimate the percentage uncertainty in this result. Justify your answer.

Figure 10 shows an arrangement of two vertical coils. Four experiments are done using this arrangement.

Coil 1 and coil 2 are identical and have a radius r. The coils are separated by a distance r and have a common axis PR. Q is at the centre of coil 1.

The four different experiments investigate how B_H varies with x, the displacement of the sensor from Q along PR.

In experiment 1, the current in coil 1 is 225 mA and the current in coil 2 is zero.

In experiment 2, the current in coil 1 is zero and the current in coil 2 is 225 mA.

Figure 11 shows the results of experiment 1 and experiment 2.

During experiment 1, B_H is measured with the sensor at Q. The sensor is then moved along PR until the value of B_H is halved. The distance from Q to the sensor is x_0.5

In experiment 3, the current in both coils is 225 mA so that the magnetic fields produced by coil 1 and coil 2 are combined.

The resultant B_H has a constant maximum value in the region between x = and

Deduce, in mT, the value of B_H in this region.

State two characteristics of the magnetic field lines in this region.

In experiment 4, the current in coil 2 is reversed so that the direction of the magnetic field produced by coil 2 is also reversed. The magnitudes of the currents in coil 1 and coil 2 are still 225 mA.

Sketch a graph to show how B_H varies between x = 0 and x = r. The x-axis has been provided for you.

Your graph should include numerical values on your B_H axis that correspond to x = 0 and x = r.

Figure 5 shows a strip of steel of rectangular cross-section clamped at one end. The strip extends horizontally over the edge of a bench.

A mass m is suspended from the free end of the strip. This produces a vertical displacement y. A student intends to measure y with the aid of a horizontal pin fixed to the free end of the steel strip. She positions a clamped vertical ruler behind the pin, as shown in Figure 6.

Explain a procedure to avoid parallax error when judging the reading indicated by the position of the pin on the ruler. You may add detail to Figure 6 to illustrate your answer.

It can be shown that

where:

L is the distance between the free end of the unloaded strip and the blocks w is the width of the strip and is approximately 1 cm t is the thickness of the strip and is approximately 1 mm E is the Young modulus of the steel.

A student is asked to determine E using the arrangement shown in Figure 5 with the following restrictions:

• only one steel strip of approximate length 30 cm is available

• m must be made using a 50 g mass hanger and up to four additional 50 g slotted masses

• the experimental procedure must involve only one independent variable

• a graphical method must be used to get the result for E.

Explain what the student must do to determine E.

Conductive putty can easily be formed into different shapes to investigate the effect of shape on electrical resistance.

A student uses vernier callipers to measure the diameter d of a uniform cylinder made of the putty.

Suggest one problem with using callipers to make this measurement.

Table 1 shows the calliper measurements made by a student.

Show that the percentage uncertainty in d is about 2.4%. Assume that all the data are valid.

The length of the cylinder is 71 ± 2 mm.

Determine the uncertainty, in mm³, in the volume of the cylinder.

A student is given some putty to form into cylinders.

To find the resistance of a cylinder, metal discs are placed in contact with the ends of the cylinder and connected to a resistance meter.

Figure 7 shows the apparatus.

The student forms the putty into cylinders of different lengths, each of volume

The student plots the graph shown in Figure 8.

Determine ρ. State an appropriate SI unit for your answer.

Figure 8

Figure 9 shows air trapped in a vertical cylinder by a valve and a piston P. The valve remains closed throughout the experiment.

A mass is placed on top of P. P moves downwards and the volume of the trapped air decreases. There are no air leaks and there is no friction between the cylinder and P.

The vertical distance y between the end of P and the closed end of the cylinder is measured. Additional masses are used to find out how y depends on the total mass M placed on top of P. Figure 10 shows a graph of these data.

Show that y is not inversely proportional to M. Use data points from Figure 10.

Figure 10

The masses are removed and the cylinder is inverted. P moves downwards without friction before coming to rest, as shown in Figure 11.

Explain why P does not fall out of the cylinder unless the valve is opened.

The mass of P is 0.350 kg.

Deduce y when the cylinder is in the inverted position shown in Figure 11.

Draw a line of best fit on Figure 10 to arrive at your answer.

Figure 12 shows apparatus used in schools to investigate Boyle’s law.

A fixed mass of air is trapped above some coloured oil inside a glass tube, closed at the top. A pump applies pressure to the oil and the air. The trapped air is compressed and its pressure p is read from the pressure gauge.

A scale, marked in 0.2 cm³ intervals, is used to measure the volume V of the air. A student says that the reading for V shown in Figure 12 is 35.4 cm³.

State:

• the error the student has made

• the correct reading, in cm³, of the volume.

Figure 13 shows data obtained using the apparatus in Figure 12.

Explain why the gradient of the graph in Figure 13 confirms that the air obeys Boyle’s law.

The largest pressure that can be read from the pressure gauge is 3.4 × 10⁵ Pa.

Determine, using Figure 13, the volume V corresponding to this pressure.

State one property of the air that must not change during the experiment. Go on to suggest how this can be achieved.

An analogue voltmeter has a resistance that is much less than that of a modern digital voltmeter. Analogue meters can be damaged if the full-scale reading is exceeded. Figure 6 shows a dual-range analogue voltmeter with a zero error.

The voltmeter is set to the more sensitive range and then used in a circuit.

What is the potential difference (pd) between the terminals of the voltmeter when a full-scale reading is indicated?

Tick () one box.

Explain the use of the mirror when reading the meter.

A student corrects the zero error on the meter and then assembles the circuit shown in Figure 7. The capacitance of the capacitor C is not known.

The output pd of the power supply is set to zero. The student connects the flying lead to socket X and adjusts the output pd until the voltmeter reading is full scale (15 V). She disconnects the flying lead from socket X so that C discharges through the voltmeter.

She measures the time T ½ for the voltmeter reading V to fall from 10 V to 5 V.

She repeats this process several times.

Table 1 shows the student’s results, none of which is anomalous.

Determine the percentage uncertainty in T ½.

Show that the time constant for the discharge circuit is about 17 s.

The student thinks that the time constant of the circuit in Figure 7 is directly proportional to the range of the meter. To test her theory, she repeats the experiment with the voltmeter set to the 3 V range. She expects T ½ to be about 2.5 s.

Explain:

• what the student should do, before connecting capacitor C to the 0 V and 3 V sockets, to avoid exceeding the full-scale reading on the voltmeter

• how she should develop her procedure to get an accurate result for the time constant

• how she should use her result to check whether her theory is correct.

The student wants to find the resistance of the voltmeter when it is set to the 15 V range. She replaces C with an 820 µF capacitor and charges it to 15 V. She discharges the capacitor through the voltmeter, starting a stopwatch when V is 14 V. She records the stopwatch reading t at other values of V as the capacitor discharges.

Table 2 shows her results.

Suggest two reasons why the student selected the values of V shown in Table 2. Explain each of your answers.

Figure 8 shows a graph of the experimental data.

Show, using Figure 8, that the resistance of the voltmeter is about 16 kΩ.

Determine the current in the voltmeter at t = 10 s.

A simple pendulum performs oscillations of period T in a vertical plane.

Figure 1 shows views of the pendulum at the equilibrium position and at the instant of release. Figure 1 also shows a rectangular card marked with a vertical line.

The card can be used as a fiducial mark to reduce uncertainty in the measurement of T.

Annotate Figure 1 to show a suitable position for the fiducial mark. Explain why you chose this position.

The period of the pendulum is constant for small-amplitude oscillations. Figure 2 shows an arrangement used to determine the maximum amplitude that can be considered to be small, by investigating how T varies with amplitude.

Describe a suitable procedure to determine A_R, the amplitude of the pendulum as it is released. You may add detail to Figure 2 to illustrate your answer.

Figure 3 shows some of the results of the experiment.

Estimate, using Figure 3, the expected percentage increase in T when A_R increases from 0.35 m to 0.70 m. Show your working.

In another experiment the pendulum is released from a fixed amplitude. The amplitudes A_n of successive oscillations are recorded, where n = 1, 2, 3, 4, 5....

Table 1 shows six sets of readings for the amplitude A₅.

Determine the result that should be recorded for A₅. Go on to calculate the percentage uncertainty in this result.

Table 2 shows results for A_n and the corresponding value of ln(A_n / m) for certain values of n.

Complete Table 2.

Plot on Figure 4 a graph of ln(A_n / m) against n.

It can be shown that

where A₀ is the amplitude of release of the pendulum δ is a constant called the damping factor.

Explain how to find δ from your graph. You are not required to determine δ.

Figure 5 shows apparatus used to investigate the bending of a beam.

The beam is placed horizontally on rigid supports. The distance L between the supports is 80 cm.

A travelling microscope is positioned above the midpoint of the beam and focused on the upper surface.

Figure 6 shows an enlarged view of both parts of the vernier scale.

The smallest division on the fixed part of the scale is 1 mm.

What is the value of the vernier reading R₀ in mm? Tick (✓) one box.

Figure 7 shows the beam bending when a hanger of mass 0.050 kg is suspended from the midpoint.

The microscope is refocused on the upper surface and the new vernier reading R is recorded. The vertical deflection s of the beam is equal to (R – R₀).

The total mass m suspended from the beam is increased in steps of 0.050 kg. A value of s is recorded for each m up to a value of m = 0.450 kg. Further values of s are then recorded as m is decreased in 0.050 kg steps until m is zero.

Student A performs the experiment and observes that values of s during unloading are sometimes different from the corresponding values for loading.

State the type of error that causes the differences student A observes.

Student B performs the experiment using a thinner beam but with the same width and made from the same material as before.

Discuss one possible advantage and one possible disadvantage of using the thinner beam.

Figure 8 shows the best-fit line produced using the data collected by student A.

where E is the Young modulus of the material of the beam and η is a constant.

Deduce in s⁻² the order of magnitude of η.

E = 1.14 GPa

Student C performs a different experiment using the same apparatus shown in Figure 5 on page 10. A mass M is suspended from the midpoint of the beam. The vertical deflection s of the beam is measured for different values of L.

Figure 9 shows a graph of the results for this experiment.

Figure 9 shows that log₁₀(s / m) varies linearly with log₁₀(L / m).

State what this shows about the mathematical relationship between s and L. You do not need to do a calculation.

Deduce, using Figure 9, the value of s when L = 80 cm.

Determine M using Figure 8.

Figure 1 shows a sealed radioactive source used in schools and colleges.

State two safety procedures to reduce risk when using this type of source.

Safety procedure 1

Safety procedure 2

226²²⁶Ra A sealed source contains radium- (₈₈).

²²⁶₈₈Ra decays by emitting and ^– particles to produce²⁰⁶₈₂Pb which is stable.

Figure 2 is a graph of neutron number N against proton number Z, showing the different ways that²²⁶₈₈Ra can decay into²⁰⁶₈₂Pb.

Points A to M represent all the unstable nuclei that may be formed as²²⁶₈₈Ra decays into²⁰⁶₈₂Pb.

Determine the number of routes by which B can change into K.

Identify which of the nuclei A to M are common to all the possible ways that²²⁶₈₈Ra

The sealed source emits γ radiation in addition to and ^– particles. A student uses the sealed source to investigate the inverse-square law for γ radiation. The student begins by making measurements to find the count rate A_b for the background radiation.

State and explain procedures

to eliminate systematic error in the measurements used to find A_b

to reduce the percentage uncertainty in A_b.

Figure 3 shows an aluminium absorber placed between the sealed source and a radiation detector. This is to make sure that only γ radiation from the source reaches the detector.

The sealed source emits:

• particles with energy E_k between 3.8 MeV and 7.8 MeV

• ^– particles with energy E_k between zero and 5.5 MeV.

Figure 4 shows how the range of ^– particles in aluminium depends on E_k.

Deduce the minimum thickness of the aluminium absorber that should be used in the experiment.

Ionisation takes place inside the detector. The effective distance travelled by radiation from the source is (d + e). The distance e, shown in Figure 5, cannot be measured directly.

From the inverse-square law for radiation, it can be shown that

where A is the count rate, corrected for background radiation k is a constant. The student plots the graph of d against shown in Figure 6.

Deduce k using Figure 6. Explain your reasoning. Give a suitable unit for your result.

Determine e using Figure 6.

Figure 7 shows apparatus used to investigate the rate at which water flows through a horizontal cylindrical tube T of internal diameter d and length L.

The apparatus ensures that the water level in the can is at a constant height h above the centre of T.

Water flows out of T at a steady rate.

The volume flow rate through T is Q, where Q is in m³ s^–1. A student wants to measure Q as water flows through T.

Outline a procedure the student should follow to measure Q. Include in your answer

the measuring instruments used

how uncertainty in the measurements can be reduced.

It can be shown that

During the experiment the temperature is kept constant. Q is found for different values of h and a graph of these data is plotted, with Q on the vertical axis. The percentage uncertainty in the gradient of the graph is 6.4%.

The dimensions of tube T are measured and the uncertainties in these data are calculated.

The percentage uncertainty

In a different experiment, the horizontal tube T is connected to a vertical glass tube. Marks have been made at regular intervals on the glass tube. The student measures and records the vertical distance y between each of the marks and the centre of T.

She seals the open end of T and fills the glass tube with water, as shown in Figure 9

T is opened and water flows into a beaker. When the water level falls to the highest mark on the tube, she starts a stopwatch. She records the time t for the water to reach each of the other marks.

Explain how the student could check that the glass tube was vertical. You may wish to add detail to Figure 9 to illustrate your answer.

Figure 10 shows part of the graph drawn from the student’s data.

It can be shown that y decreases exponentially with t.

T₁₂ is the time for y to decrease by 50%, as shown in Figure 11.

The apparatus is adjusted so that the glass tube is inclined at 30 to the horizontal tube T, as shown in Figure 12.

The student measures and records the new values of y, the mean vertical distance between each of the marks and the centre of T. She then carries out the experiment as before, recording new values of t corresponding to each new value of y.

Draw a line on Figure 13 to show the graph produced using the modified apparatus. The dashed line is the original graph when the glass tube was vertical as shown in Figure 9.

A stationary wave is formed on a stretched wire.

Figure 14 shows the wire, fixed at one end, supported by two bridges and passing over a pulley.

A 0.500 kg mass is attached to the free end of the wire. A uniform horizontal magnetic field is applied perpendicular to the wire between the bridges. A signal generator is connected to each end of the wire.

The oscilloscope shown is used to determine the frequency of the output of the signal generator. The wire oscillates because the alternating current in the wire interacts with the magnetic field.

Figure 15 shows the first harmonic stationary wave produced when the distance x between the bridges is adjusted.

The output potential difference (pd) of the signal generator is displayed on the oscilloscope, as shown in Figure 16.

The time-base setting of the oscilloscope is 10 ms cm¹.

Determine f, the frequency of the alternating pd.

A metre ruler is placed next to the bridges supporting the wire, as shown in Figure 17

Determine c.

Determine, in kg m¹, the mass per unit length of the wire.

A student uses digital vernier callipers to measure the diameter of a cylindrical metal rod. The student places the rod between the jaws of the callipers and records the reading indicated. Without pressing the zero button, the student removes the rod and closes the jaws.

Figure 18 shows the calliper readings in millimetres, before and after the jaws are closed.

Calculate the diameter d of the rod.

Describe relevant procedures to limit the effect of random error in the result for d.

Determine the density of the rod. The mass per unit length of the rod is 3.54 10³ kg m¹.

A student carries out an experiment to determine the resistivity of a metal wire. She determines the resistance from measurements of potential difference between the ends of the wire and the corresponding current. She measures the length of the wire with a ruler and the diameter of the wire using a micrometer. Each measurement is made with an uncertainty of 1%

Which measurement gives the largest uncertainty in the calculated value of the resistivity?

This question is about using a digital balance to investigate the force on a wire placed in a magnetic field when there is an electric current in the wire.

A student carries out the procedure shown in Figure 1 and Figure 2. A metre ruler is pivoted at the 1.0 cm mark and a prism is placed on a digital balance. The free end of the ruler is raised and the balance is turned on and then set to zero, as shown in Figure 1.

The ruler is then supported by the prism with the apex of the prism at the 30.0 cm mark as shown in Figure 2. The height of the pivot is adjusted so that the ruler is horizontal.

Deduce the mass of the ruler. State one assumption you make.

The student attaches a uniform wire to the upper edge of the ruler, as shown in Figure 3.

The ends of the wire are connected to terminal blocks P and Q which are fixed firmly to the bench. A power supply and an ammeter are connected between P and Q.

These modifications cause the balance reading to increase slightly.

A horizontal uniform magnetic field is applied, perpendicular to the wire, between the 85 cm and 90 cm marks, as shown in the close-up diagram in Figure 3.

The balance reading M is recorded for increasing values of current I. A graph of these data is shown in Figure 4.

State and explain the direction of the horizontal uniform magnetic field.

It can be shown that B, the magnitude of the magnetic flux density of the horizontal uniform magnetic field, is given by

where change in force acting on the prism per unit current in the wire L = length of the region where the magnetic field cuts through the wire.

Determine B.

The experiment is repeated with the ruler pivoted at the 99.0 cm mark. Nothing else is changed from Figure 3.

This arrangement is shown in Figure 5.

Tick () one box in row 1 and one box in row 2 of Table 1 to identify the effect, if any, on the magnitude of the forces acting on the apparatus as a certain current is passed through the wire.

Tick () one box in row 3 and one box in row 4 of Table 1 to identify the effect, if any, on the graph produced for this modified experiment compared with the graph in Figure 4.

Figure 6 shows the balance being used to measure the forces between two wires. The connections joining these wires to the power supply are not shown.

The pan of the balance moves a negligible amount during use and it supports a straight conducting wire X of horizontal length L. Terminal blocks are used to connect X into the circuit. The weight of these does not affect the balance reading. A second conducting wire Y is firmly supported a distance d above X.

Show, by adding detail to Figure 6, the wire connections that complete the circuit. The currents in X and Y must have the same magnitude and be in the directions indicated.

The vertical force F on wire X due to the magnetic field produced by wire Y is given by

where k is a constant d is the perpendicular distance between X and Y I is the current in the wires and L is the horizontal length of wire X.

A student wants to measure k using the arrangement in Figure 6.

The student is told that the following restrictions must apply:

• L is fixed

• I must not exceed 5.0 A

• the result for k must be obtained using a graphical method

• the experimental procedure must involve only one independent variable.

Explain what the student could do to find k.

This question is about an experiment with a retractable steel tape measure.

The tape measure is placed at the edge of the bench and about 1 m of the steel tape is extended so that it overhangs the bench.

The tape is then locked in this position to stop it from retracting.

A student measures the dimensions x and y, the horizontal and vertical displacements of the free end of the tape, as shown in Figure 7.

Describe a suitable procedure the student could use to measure y. You may add detail to Figure 7 to illustrate your answer.

By changing the extension of the tape, the student obtains further values of x and y.

These data are shown in Table 2.

Suggest why the student chose to make all measurements of x greater than 70 cm

The data from the experiment suggest that y = Axⁿ where n is an integer and A is a constant.

These data are used to plot the graph in Figure 8.

Determine n using Figure 8.

Figure 8

Explain how the numerical value of A can be obtained from Figure 8.

Estimate the order of magnitude of A. You should use data for x and y from any one row in Table 2 on page 11. Give your answer with an appropriate unit.

This question is about an experiment to estimate absolute zero.

Figures 9a to 9d show the stages in the procedure carried out by a student.

An empty flask fitted with a tube and an open valve is placed in water bath H containing hot water. The air inside the flask is allowed to come into thermal equilibrium with the water. The valve is then closed, trapping a certain volume of air, as shown in Figure 9a.

The flask is inverted and placed in water bath C in which the water is at room temperature. The air inside the flask is again allowed to come into thermal equilibrium with the water, as shown in Figure 9b.

The valve is opened and some water enters the flask, as shown in Figure 9c.

The depth of the inverted flask is adjusted until the level of water inside the flask is the same as the level in the water bath. The valve is then closed, trapping the air and the water inside the flask, as shown in Figure 9d.

Explain why the volume of the air in the flask in Figure 9c is less than the volume of the air in the flask in Figure 9d.

Explain why Charles’s Law can be applied to compare the air in the flask in Figure 9a with the air in the flask in Figure 9d.

The flask is removed from water bath C and the valve and stopper are removed. The volume of the water in the flask is V₁ The flask is then completely refilled with water and the valve and stopper replaced. The volume of the water now in the flask is V₂ The volumes V₁ and V₂ are shown by the shaded parts in Figure 10.

Explain how V₁ and V₂ can be determined.

In your answer you should

• identify a suitable measuring instrument

• explain a suitable procedure to eliminate possible systematic error.

Plot on Figure 11 points to show the volume V and the temperature θ of the air in the flask when

• the flask is as shown in Figure 9a

• the flask is as shown in Figure 9d.

The temperature of the hot water bath is 86 C

Add a best fit line to your graph in Figure 11 to show how V should vary with θ according to Charles’s Law.

Determine the value of absolute zero in C using your graph in Figure 11.

value of absolute zero =

A microwave transmitter T and a receiver R are arranged on a line marked on the bench.

A metal sheet M is placed on the marked line perpendicular to the bench surface.

Figure 1 shows side and plan views of the arrangement. The circuit connected to T and the ammeter connected to R are only shown in the plan view.

The distance y between T and R is recorded.

T is switched on and the output from T is adjusted so a reading is produced on the ammeter as shown in Figure 2.

M is kept parallel to the marked line and moved slowly away as shown in Figure 3

The reading decreases to a minimum reading which is not zero. The perpendicular distance x between the marked line and M is recorded.

The ammeter reading depends on the superposition of waves travelling directly to R and other waves that reach R after reflection from M.

State the phase difference between the sets of waves superposing at R when the ammeter reading is a minimum. Give a suitable unit with your answer.

Explain why the minimum reading is not zero when the distance x is measured.

When M is moved further away the reading increases to a maximum then decreases to a minimum.

At the first minimum position, a student labels the minimum n = 1 and records the value of x. The next minimum position is labelled n = 2 and the new value of x is recorded. Several positions of maxima and minima are produced.

Describe a procedure that the student could use to make sure that M is parallel to the marked line before measuring each value of x. You may wish to include a sketch with your answer.

It can be shown that

The student plots the graph shown in Figure 4.

and adds error bars to the graph.

Determine

the maximum gradient G_max of a line that passes through all the error bars

the minimum gradient G_min of a line that passes through all the error bars.

Explain how the graph in Figure 4 can be used to obtain the value of y. You are not required to determine y.

Suppose that the data for n = 13 had not been plotted on Figure 4.

Add a tick () in each row of Table 1 to identify the effect, if any, on the results

A block is used to raise one end of the track. A bumper fitted with a rubber cord is attached at the lower end of the track. The air track has a length of 2 m and there is a scale with major divisions marked in centimetres along the side; the zero of the scale is at the lower end, as shown in Figure 13.

A glider is placed in contact with the rubber cord on the bumper at the lower end of the track. The position of the glider relative to the fixed scale can be determined using Figure 14.

The glider is then moved to the position shown in Figure 15.

Figure 15

The air supply to the track is turned on and the glider is released. The glider accelerates down the track, strikes the rubber cord on the bumper and rebounds back up the track. The glider is allowed to bounce off the rubber band 20 times before it is stopped.

A student reads and records the highest position p of the glider after each rebound n. Some of the student’s data are shown in Table 2. Additional columns have been provided to allow you to complete question 03.2 and question 03.3.

The value of p corresponding to n = 0 is the glider’s initial position at the top of the track.

Deduce this value of p using Figure 13 and Figure 15. Write this result in Table 2.

As it travels from the lower end of the track to each position p the glider moves through a distance x.

Deduce x for all the values of n using Figure 14. Write these results in Table 2.

Plot on Figure 16 a graph of ln (x /cm) against n.

Record your values of ln (x /cm) in Table 2.

Explain why the graph you plotted confirms that x decreases exponentially with n.

Figure 16

Determine, using your graph in Figure 16, the value of x when n is 20.

Describe and explain two procedures the student should take to reduce uncertainty in the measurements of p.