Uncertainties & Errors (DP IB Physics: HL)

A vernier calliper is used to measure the length of a piece of copper wire.

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 the 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. Give the range of the uncertainty of this extension.

Tensile strain is a measure of the deformation of an object and is defined as the ratio between the extension of the wire and its original length.

                                           Tensile Strain, ε = 

Deduce the tensile strain of the copper wire and its percentage uncertainty.

State two ways to reduce the systematic error in this experiment.

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

The table shows the period T recorded ten times.

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 a graph.

Explain how the graph 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:

                                                                 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.

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.

A student performs an experiment to find the acceleration due to gravity. A spherical object falling freely through measured vertical distances s for a time t. The experiment is repeated in a lab and the time is measured electronically.

Plot the data on the graph below, including error bars and a line of best fit.

Calculate the value of g for this experiment.

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 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 to reduce systematic errors in the measurement of x. Use a sketch to aid 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.

A relationship between x and y against n is shown on the graph. The wavelength λ is the gradient of the graph.

Determine the maximum and minimum possible values of λ.

Determine:

   (i) The value of λ

   (ii) The percentage uncertainty in the value of λ.

Another student conducted a similar experiment but determined the uncertainty in the relationship of x and y to be 0.010 m for each term.

Explain the effect this would have on the uncertainty in λ.

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

where C is the count rate of the sample at time t, C₀ is the initial count rate 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 results in the table show the total count for exactly 5 minutes, repeated at 15 minute intervals.

The uncertainty in the count rate, C, is given by

                                                           ΔC = ±

Calculate the uncertainty in each value of ln C.

Draw a line of best fit and error bars for each point on the graph.

The activity of the sample λ = –

Calculate the activity of the sample and its percentage 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.  

Use the relationship 

                                                            ln(x) = y so x = e^y

to estimate the percentage uncertainty in this total count and explain the advantage of using a larger time.

One method to determine the acceleration of free fall g involves measuring the time period of a simple pendulum T. It is related to the length of the pendulum l by the equation

In this method, l was found to be (0.500 ± 0.001) m while the period T was measured to be (1.42 ± 0.02) s. 

Based on these measurements, determine the value of g and its absolute uncertainty. Give your final answer to an appropriate degree of precision. 

Another method to determine the acceleration of free fall involves timing the descent of a small metal ball bearing, released vertically via an electromagnetic trapdoor. In one particular trial, the displacement of the ball bearing s is measured as (266 ± 1) cm and the time measured t is (0.740 ± 0.005) s. 

Determine the value of g using this method, and its absolute uncertainty. Give your final answer to an appropriate degree of precision. 

The length l of a simple pendulum is increased by 6%.

Determine the fractional increase in the pendulum's period, T. 

You may use the relationship between period T and length l as: 

where g is the acceleration of free fall. 

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

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 biggest 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:

P = eσAT⁴

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 part (b).