Processing Uncertainties (DP IB Physics: SL): Exam Questions

Two students, A and B, investigate the extension of a wire when put under various loads. The wire is clamped horizontally at one end and a weight is attached to the other end and allowed to hang vertically. For each value of , the group measures the vertical length of the wire.

Student A measures using vernier callipers and records their data in the following table.

The vernier calliper has a positive zero error of 0.10 mm.

Indicate, in the table, the corrected readings for for each value of .

Student A wants to determine the extension of the wire for each value of . The vernier calliper can measure to the nearest 0.02 mm.

Using the data, calculate the extension of the wire when = 1.50 N, and give the absolute uncertainty.

Student B measures using a ruler and records their data in the following table.

The ruler can measure to the nearest 1.0 mm.

Calculate the fractional uncertainty in using a ruler when = 2.50 N. Give the final value with its fractional uncertainty.

Student B obtains the following repeated readings for for one value of .

Student B asks Student A to help analyse the data.

Student A quotes the percentage uncertainty in as 33%.

Student B quotes the percentage uncertainty in as 10.2%.

Discuss the values that the students have quoted.

In an experiment to measure the acceleration due to gravity , a student carries out a procedure using a simple pendulum.

The student suggests that the period of oscillation is related to the length of the pendulum by the equation

The student obtains the following repeated readings for .

Determine the mean value of and its percentage uncertainty.

For a different value of , the student measures the time for the pendulum to complete 20 oscillations to be (18.4 ± 0.1) s.

The percentage uncertainty in is determined to be 1.8%.

Calculate the percentage uncertainty in .

The student plots a graph of the variation of with .

Explain how the graph indicates the presence of systematic and random errors in the data.

The period of oscillation for a mass hanging on a spring performing simple harmonic motion is given by

where is the spring constant of the spring.

The fractional uncertainty in is and the fractional uncertainty in is .

Determine the fractional uncertainty in in terms of and .

An object falls off a cliff of height h above the ground. It 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 fractional uncertainty.

The only instrument used in this experiment was a stopwatch.

(i) Discuss one possible source of systematic error and one possible source of random error in this investigation.

[2]

(ii) Explain how these errors could influence the value of acceleration of free-fall calculated in (a).

[2]

The diagram shows the side and plan views of a microwave transmitter M_T and a receiver M_R arranged on a line marked on the bench.

The circuit connected to M_T and the ammeter connected to M_R are only shown in the plan view.

The distance  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  between the marked line and M is recorded.

Describe one method to reduce systematic errors in the measurement of .

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

A relationship between and against is shown on the graph. The wavelength is the gradient of the graph.

Determine the maximum and minimum possible values of .

Calculate:

(i)

[2]

(ii) the percentage uncertainty in .

[2]

The relationship between the period  and length  of a simple pendulum is: 

where  is the acceleration of free fall.

Determine the fractional increase in if is increased by 6%.

The time period T of a pendulum is also related to the amplitude of oscillations θ. Measurements are taken and a graph is obtained showing the variation of with angular amplitude θ, where T₀ is the period for small amplitude oscillations:

Use the information from the graph to

(i) deduce the condition for the time period T to be considered independent of angular amplitude θ.

[2]

(ii) determine the maximum value of θ for which T is independent of θ.

[2]

Typically, using a simple pendulum to determine the acceleration of free fall g involves measuring the periodic time T and the pendulum length l. 

State and explain which piece of measuring equipment is likely to have the largest impact on the accuracy of the value determined for g. 

An experiment is designed to explore the relationship between the temperature of a ball T and the maximum height to which it bounces h. 

The ball is submerged in a beaker of water until thermal equilibrium is reached. The ball is then dropped from a constant height and the height of the first bounce is measured. This is repeated for different temperatures. The results are shown in the graph, which shows the variation of the mean maximum height h_mean with temperature T:

Compare and contrast the uncertainties in the values of h_mean and T. 

The experimenter hypothesises, from their results, that h_mean is proportional to T². 

Suggest how the experimenter could use two points from the graph to validate this hypothesis. 

State and explain whether two points from the graph can confirm the experimenter's hypothesis.

It is known that 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.

In an experiment to determine the emissivity e of a circular surface of diameter d, the following measurements are taken: 

• P = (3.0 ± 0.2) W

• d = (6.0 ± 0.1) cm

• 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. 

The power dissipated in a resistor can be investigated using a simple electrical circuit. The current in a fixed resistor, marked as 47 kΩ ± 5%, is measured to be (2.3 ± 0.1) A. 

Determine the power dissipated in this resistor with its associated uncertainty. Give your answer to an appropriate degree of precision. 

A student investigates the relationship between two variables T and B. Their results are plotted in the graph shown: 

Comment on the absolute and fractional uncertainty for a pair of data points.

The student suggests that the relationship between T and B is of the form:

where a and c are constants. To test this suggested relationship, the following graph is drawn:

Describe a method that would determine the value of c and its uncertainty. 

Comment on the student's suggestion from (b).