Limitation of Physical Measurements (AQA A Level Physics)

Figure 1 shows the reading on a micrometer screw gauge.

Figure 1

Determine the full reading on the micrometer with its absolute uncertainty. 

A student measures and records 10 repeat readings of the diameter of a wire using a micrometer screw gauge in mm shown in Table 1.

     Table 1

 Determine the mean diameter of the wire and its absolute uncertainty.

Determine the percentage uncertainty in the result the student obtains for the diameter of the wire.

 Give your answer to an appropriate number of significant figures.

For the experiment, the student requires the cross-sectional area of the wire.

Determine the cross-sectional area of the wire in mm² and its percentage uncertainty.

Figure 1 shows the reading on a vernier calliper.

Figure 1

Determine the full reading of the vernier calliper with its absolute uncertainty. 

The reading from part (a) is taken after a mass has been added to a copper wire of length L and the wire extends.

 The original length of the wire L­ was 14.9 ± 0.05 mm.

Calculate the extension ∆L of the copper wire after the mass has been added with its absolute uncertainty.      

Tensile strain is defined as the ratio between the extension of the wire and its original length.

Hence, or otherwise, determine the tensile strain of the copper wire and its percentage uncertainty. 

Suggest one way to improve the experiment to make the value of the wire extension more accurate.

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 length of the pendulum L and by the equation:

             T = 2π

They measure length L to be 11.2 cm and measure the period T ten times. The results are recorded in Table 1

 Table 1

Determine the mean period of oscillation and its percentage uncertainty.

State one way the student could reduce the uncertainty in their value of T.

The student estimates the uncertainty on the measurement of L to be ± 15 mm.

Hence, or otherwise, use your answer from part (a) to determine the percentage uncertainty in the value of g.

Hence, or otherwise, use your answer from part (c) to determine the value and absolute uncertainty in the value of g.

Give your answer to an appropriate unit and number of significant figures.

Write the following currents in order of decreasing percentage uncertainty:

   4.1 ± 0.2 A,     5 ± 1 mA,        7.30 ± 0.23 A,      0.5 ± 0.05 mA

A circuit is set up to measure the resistance R of a resistor. The potential difference (p.d) V across the resistor and the current I are measured.

 The readings for the p. d V and the corresponding current I are obtained. These are shown in Figure 1.

Figure 1

Explain how Figure 1 indicates that the readings are subject to a systematic uncertainty and random uncertainties.

State one way a systematic error could have occurred in this experiment and how this type of error can be fixed.

In another experiment, the resistance of the resistor R is determined using the following data:     

         Current, I = 0.74 ± 0.01 A 

      Potential difference, V = 6.5 ± 0.2 V

Calculate the value of R, together with its percentage uncertainty. Give your answer to an appropriate number of significant figures.

A student has a diffraction grating that is marked 2.9 × 10³ lines per m.

Calculate the percentage uncertainty in the number of lines per metre suggested by this marking. 

Give your answer to an appropriate number of significant figures.

Determine the grating spacing and its absolute uncertainty in mm.

Figure 1 shows part of another diffraction grating. A scale is given below.

Figure 1

Use Figure 1 to calculate the grating spacing of this diffraction grating. 

State an appropriate absolute uncertainty and calculate the percentage uncertainty.

Hence, or otherwise, calculate the number of lines per metre on the grating.

An object falls off a cliff which is at a height, h, above the ground. The object takes 13.8 seconds to hit the ground.

 It is estimated that there is a percentage uncertainty of ±5% in measuring this time interval. A guidebook of the local area states the height of the cliff is 940 ± 10 m.

Calculate the acceleration of free-fall of the object and its absolute uncertainty to an appropriate number of significant figures. 

The only instrument used in this experiment was a stopwatch.

Suggest one possible source of systematic error and one source of random error in this investigation and explain how these errors could influence the value of acceleration of free-fall of the object determined in part (a).

The experiment is repeated in a lab and the time is measured electronically. A student performs an experiment to find the acceleration due to gravity.

 The time t for a spherical object to fall freely through measured vertical distances s is measured. The results are shown in Table 1. 

Table 1

 Complete the missing data in Table 1 and plot the data on Figure 1 below, including error bars and a line of best fit.

Figure 1

The uncertainty in each value of s is ± 0.002 m.

Hence, or otherwise, calculate the value of g for this experiment and its percentage uncertainty.

A microwave transmitter M_T and a receiver M_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 M_T and the ammeter connected to M_R are only shown in the plan view.

Figure 1

The distance y between M_T and M_R is recorded.

 M_T is switched on and the output from M_T is adjusted so a reading is produced on the ammeter.

 M is kept parallel to the marked line and moved slowly away. The perpendicular distance x between the marked line and M is recorded.

Describe one method of reducing systematic errors in the measurement of x. 

 You may wish to include a sketch with your answer.

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.

It can be shown that the relationship between positions of the maxima and minima, n, the wavelength of the microwaves, λ, and the distances x and y, as defined in Figure 1, is equal to:

          nλ = 

 The student plots the graph shown in Figure 2.

 The student estimates the uncertainty in each value of  to be 0.025 m and adds error bars to the graph.

Figure 2

Using Figure 2, determine the maximum and minimum possible values of λ.

Hence, determine λ and the percentage uncertainty in the value of λ.

Another student conducted a similar experiment but determined the uncertainty in each value of  to be 0.010 m.

Using Figure 2, explain the effect this would have on the uncertainty in λ.

An object of mass 1.000 g is placed on four different balances. For each balance the reading is taken five times, as shown in Table 1. 

Table 1

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

In an experiment to measure the density of steel, a steel sphere was used. The diameter of the steel sphere was found to be 0.60 cm ± 0.01 mm. 

The object used in part (a) was the steel sphere. The mass of the sphere was measured using balance 2 shown in Table 1.

Determine the density of the sphere and its associated percentage uncertainty.

The experimenter repeats the experiment to measure the density of steel, but this time with a cylindrical container made from steel.

 The volume of the cylinder was found by measuring its diameter and height to within 0.01 and 0.03 fractional uncertainty respectively.

 The measurements obtained were as follows: 

Diameter of the cylinder = 2.2 cm

Height of the cylinder = 25.0 cm

Determine the density of the cylinder and its associated percentage uncertainty. 

The true value of the density of the steel used in both the cylinder and the sphere was found to be 8.10 g cm^–3. 

A student assumes that the percentage difference between the density of the steel cylinder and the true value is due only to the uncertainty in the volume, and the uncertainty in the mass.

Determine the validity of this assumption.

The period of oscillation of a pendulum is given by the equation: 

            T  = 2π

Where 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 a graph, as shown in Figure 1.

Figure 1

 The graph was expected to pass through the origin.

Suggest a plausible explanation for the non-zero intercept.

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

               T  = 2π

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.

The spring constant of a spring may be determined by finding the extension of the spring and the load applied using the apparatus shown in Figure 2.

Readings of the position of the lower end of the spring are made using a metre rule.

Evaluate the precision and accuracy of this experiment.

In your answer you should make reference to:

• Possible sources of random and systematic errors

• Methods for keeping errors in the readings to a minimum

The decay of a radioactive substance can be represented by the equation

where A is the activity of the sample at time t, A₀ is the initial activity at time t = 0 and λ is the decay constant.

 The half-life, t_½ of the radioactive substance is given by

 An experiment was performed to determine the half-life of a radioactive substance which was a beta emitter. The radioactive source was placed close to a detector.

 The total count for exactly 5 minutes was recorded. This was repeated at 15 minute intervals. The results are shown in Table 1.

 Table 1

It can be shown that the uncertainty in the corrected count rate, C, is given by 

            ΔC = ± √C

Use this information to calculate the uncertainty in each value of ln C. 

On Figure 1, draw a line of best fit and error bars for each point.

Hence, determine the half-life of the sample and its associated absolute uncertainty.

Another student performed the same experiment with identical equipment but took total counts over a 1-minute period rather than a 5-minute period. The total count, C, at 140 minutes was equal to 54 counts.

Estimate the percentage uncertainty in this total count, and hence explain the advantage of using a larger time.