1.2 Uncertainties & Errors

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.

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%. 

Measurements of time periods for different lengths of pendula were taken using a stopwatch and plotted on a graph.

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.

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.

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.

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.

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.

   (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.

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 = ±

(a)        Calculate the uncertainty in each value of ln C.

The activity of the sample λ = –

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.  

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