Errors & Uncertainties (Cambridge (CIE) A Level Physics): Exam Questions

The speed v of a transverse wave on a uniform string is given by the expression

where T is the tension in the string, l is its length and m is its mass.

An experiment is performed to determine the speed v of the wave. The measurements are shown in Table 1.1.

Table 1.1

Use the data in Table 1.1 to calculate the speed v.

v = ................................................. m s^–1 

Use your answer in (a) and the data in Table 1.1 to calculate the absolute uncertainty in the value of v.

absolute uncertainty = ................................................. m s^–1 

A student used the apparatus shown to investigate the time taken for a hollow cylinder to roll down a ramp as shown in Fig. 1.1.

State the most appropriate instrument for the measurement of the following, and describe how they can ensure the measurements are as accurate as possible 

(i) the heights h₁ and h₂

[2]

(ii) the time taken for the cylinder to roll down the ramp

[2]

The student wants to find the difference in height Δh between the start line and finish line

The heights are recorded as

= 67.5 ± 0.1 cm

= 8.6 ± 0.1 cm

Determine the change in height Δh, quoting your answer with its uncertainty

 Δh = …………………… ± …………………… cm 

The student placed the cylinder on the start line and released it. She measured the time t for the cylinder to roll to the finish line. She repeated the measurements several times as shown in Table 1.1.

Table 1.1

(i) Use the data in Table 1.1 to calculate the mean value of t and its uncertainty.

mean value of t = ............................... ± ..........................  [2]

(ii) Comment on the accuracy and precision of the data the student collected.

[2]

(iii) Explain how reducing the value of h₁ could improve the measurement of t.

[2]

The relationship between t and the acceleration of free fall g is given by

where s is the distance along the ramp between the start line and the finish line.

The student recorded the ramp length as s = 102.0 cm ± 0.1 cm

(i) Determine a value for g.

[2]

(ii) Determine the percentage uncertainty in the value of g.

[2]

(iii) Deduce whether the value of g is accurate.

[2]

A cube made from aluminium is placed on a horizontal surface, as shown in Fig. 1.1.

The following measurements were made on the cube:

mass = 0.59 ± 0.01 kg

length of one side = 6.0 ± 0.1 cm

Calculate the pressure produced by the cube on the surface.                                                                               

(i) Calculate the percentage uncertainty in the pressure.

[2]

(ii) State the pressure, with its actual uncertainty.

pressure = .................... ± .................... Pa  [2]

Calculate the density of aluminium with its uncertainty.

Express your answer to an appropriate number of significant figures.

 density .................. ± .................. g cm⁻³  

The student uses vernier callipers to measure the length of the side of the cube. After completing the experiment, he realises that when the callipers are fully closed, the reading is not zero. 

(i) State and explain whether this introduces a random error or a systematic error in the readings of the length.

[2]

(ii) Explain why the readings are precise but not accurate.

[2]

Analogue voltmeters and cathode-ray oscilloscopes (c.r.o.) are both devices which can measure potential differences.

For the analogue voltmeter and the c.r.o., describe one example of:

(i) a systematic error that can affect both devices

[1]

(ii) a systematic error that can only affect the analogue voltmeter

[1]

(iii) a random error that can affect both devices

[1]

(iv) a random error that can only affect the analogue voltmeter.

[1]

The potential difference across a resistor is measured using the analogue voltmeter shown in Fig. 1.1.

The resistor is labelled as having a resistance of 125 Ω ± 3%.

(i) Calculate the power dissipated by the resistor.

[2]

(ii) Calculate the percentage uncertainty in the calculated power.

[2]

(iii) Determine the value of the power, with its absolute uncertainty, to an appropriate number of significant figures.

  power = ............................... ± ................................ W  [2]

A microphone connected to a cathode ray oscilloscope (c.r.o.) detects a sound wave of frequency f. The signal from the microphone is observed on the c.r.o. as shown in Fig. 1.2.

The time-base setting of the c.r.o. is 0.50 ms cm^–1. The Y-plate setting is 2.5 mV cm^–1.

Using the trace in Fig. 1.2, determine 

(i) the amplitude of the signal, in mV

 amplitude = ............................................. mV  [2]

(ii) the frequency, in Hz

frequency = .................................................... Hz  [3]

(i) Determine the percentage uncertainty in amplitude caused by reading the scale on the c.r.o.

[2]

(ii) State the amplitude with its actual uncertainty.

amplitude = ................................ ± ................................ mV  [1]

A student participates in an experiment to measure the Earth’s gravitational field strength g. This is done using a simple pendulum.

The student suggests the period of oscillation T is related to the length of the pendulum L by the equation:

Table 1.1 shows the ten repeat readings of period T.

Table 1.1

Determine the mean period of oscillation and its percentage uncertainty.

In a new experiment, the length of the pendulum L is measured with an accuracy of 1.8% and the acceleration due to free-fall g is measured with an accuracy of 1.6%. 

If the time for the pendulum to complete 20 oscillations is 18.4 s, determine the time period for one oscillation and the absolute uncertainty in this value.

Measurements of time periods for different lengths of pendula were taken using a stopwatch and plotted on the graph shown in Fig. 1.1.

Fig. 1.1

Explain how the graph in Fig. 1.1 indicates that the readings are subject to systematic and random uncertainties.

The period T for a mass m hanging on a spring performing simple harmonic motion is given by the equation:

Such a system is used to determine the spring constant k. The fractional error in the measurement of the period T is α and the fractional error in the measurement of the mass m is β.

Determine the fractional error in the calculated value of k in terms of α and β.

In an experiment to determine the emissivity of a circular surface, the diameter d of the object is measured using the vernier scale shown in Fig. 1.1.

Determine the diameter of the circular object from the reading in Fig. 1.1 and its associated uncertainty.

The energy per unit time P radiated by an object with surface area A at absolute temperature T is given by

where e is the emissivity of the object and σ is the Stefan-Boltzmann constant.

Show that the emissivity e has no unit.

As well as diameter d, the following measurements were taken: 

P = 3.0 ± 0.1 W

T = 500 ± 1 K

Determine the value of the emissivity e of the surface and its uncertainty. Give your answer to an appropriate degree of precision.

Fig. 1.1 shows a circuit designed by a student investigating the variation of potential difference with time as a capacitor charges and discharges whilst maintaining a constant current.

Fig. 1.1

The power supply has an e.m.f. of 6.0 V and negligible internal resistance. During the charging process, the maximum value of the resistance of the variable resistor is 100 kΩ.

(i) Suggest one advantage of using an analogue ammeter rather than a digital ammeter for this experiment.

[1]

(ii) Fig. 1.2 shows three analogue ammeters which can detect currents up to a maximum of 50 µA, 100 µA or 500 µA. Suggest, with a calculation, which of the analogue ammeters in Fig. 1.2 would be most suitable for the experiment.

[3]

(iii) State the uncertainty in a reading of current using the ammeter you choose in (ii).

[1]

A digital voltmeter with a three-digit display is used to measure the potential difference across the capacitor. A reading made during the charging process is shown in Fig. 1.3.

The manufacturers of the meter state that its accuracy is ±2% and ±1 digit.

(i) Determine the maximum possible value of the potential difference across the capacitor at the moment the reading in Fig. 1.3 was taken.

[2]

(ii) The reading on the voltmeter has high precision. State and explain why the reading may not be accurate.

[2]

Initially, the shorting lead is connected across the capacitor, as shown in Fig. 1.1, to fully discharge it. The timer is started at the moment the shorting lead is disconnected as the capacitor begins to charge.

The student measures the potential difference across the capacitor at 5 s intervals while adjusting the variable resistor to keep the charging current constant.

The label on the capacitor states its capacitance  is 510 μF.

Determine the number of different readings the student will be able to take before the capacitor becomes fully charged.

The charge on the capacitor is equal to 

The capacitor used in the experiment has a manufacturing tolerance of ±10%. Throughout the experiment, the charging current is maintained at the value calculated in (a).

The data from the experiment produces a straight-line graph for the variation of potential difference with time. This shows that the potential difference across the capacitor increases at a rate of 110 mV s⁻¹.

(i) Calculate the true capacitance of the capacitor.

[2]

(ii) Deduce whether the capacitor is within the manufacturer’s tolerance.

[2]

An object of mass 50.00 g is placed on four different balances. For each balance, the reading is taken five times, as shown in Table 1.1.

Table 1.1

Analyse the data and conclude which sets of readings are accurate, precise or neither.

A titanium sphere and a titanium cylinder were used in an experiment to measure the density of titanium. The diameter of the sphere was measured using a micrometer, and the diameter of the cylinder was measured using vernier callipers, as shown in Fig. 1.1.

Fig. 1.1

Using Fig. 1.1, determine

(i) the diameter of the titanium sphere, in cm, with its uncertainty.

[2]

(ii) the diameter of the titanium cylinder, in cm, with its uncertainty.

[2]

The measurements for the mass and height of the cylinder with their fractional uncertainties are shown in Table 1.2.

Table 1.2

The object used in (a) was the titanium sphere. The mass of the sphere was measured using the balance 2 shown in Table 1.1.

Determine 

(i) the density of the titanium sphere, in g cm⁻³, and its associated percentage uncertainty.

[4]

(ii) the density of the titanium cylinder, in g cm⁻³, and its associated percentage uncertainty.

[3]

The true value of the density of the titanium used in both the cylinder and the sphere was found to be 4.51 g cm^–3. 

The student assumes that the percentage difference between the density of the titanium cylinder and the true value is solely due to the uncertainty in the volume and the uncertainty in the mass. 

Assess the validity of this assumption.