1.2 Uncertainties & Errors

The measurement of a physical quantity may be subject to random errors and systematic errors.

Which statement is correct?

A stone falls from rest to the bottom of a water well of depth d. The time t taken to fall is 3.0 ± 0.3 s. The depth of the well is calculated to be 30 m using d = at². The uncertainty in a is negligible.

What is the absolute uncertainty in d?

The diagram shows an experiment to measure the speed of a small ball falling at constant speed through a clear liquid in a glass tube.

There are two marks on the tube. The top mark is positioned at 15 ± 1 mm on the adjacent rule and the lower mark at 285 ± 1 mm. The ball passes the top mark at 2.50 ± 0.02 s and passes the lower mark at 4.50 ± 0.02 s.

The constant speed of the ball is calculated to be 135 mm s^–1.

Which expression calculates the fractional uncertainty in the value of this speed?

The strain energy W of a spring is determined from its spring constant k and extension x. The spring obeys Hooke’s law and the value of W is calculated using the equation shown.

                                                                 W = kx²

The spring constant k is 200 ± 1 N m^–1 and the extension x is 0.040 ± 0.004 m.

What is the percentage uncertainty in the calculated value of W?

In an experiment investigating the electrolysis of copper, a student sets out to find the electrochemical equivalent, Z.

The electrochemical equivalent of a substance is the amount of substance deposited on a cathode per Coulomb of charge.

This can be determined using the equation:

                                             Z = 

Where:

• Mass of cathode before passing current, = (54.39 ± 0.01) × 10^–3 kg
• Mass of cathode after passing current, = (52.06 ± 0.01) × 10^–3 kg
• Current, = 3.00 ± 1 A
• Time, t = 4800 ± 100 s

What is the largest possible value of Z from these readings?

The sides of a square are measured to be 8.0 ± 0.2 cm.

Which of the following gives the area of the square and its uncertainty?

In an experiment, a radio-controlled car takes 1.50 ± 0.05 s to travel 30.0 ± 0.1 m.

What is the car’s average speed and the uncertainty in this value?

The diagram shows a thermometer reading of a liquid's temperature, before and after heating. 

What is the best estimate for the temperature increase of the liquid?

A student collects values of the time period, T, of a pendulum at different lengths of string, L. They want to investigate the relationship:

                                                                T = 2π 

They plot the values on a graph along with the error bars associated with each point, as shown in the diagram.

What is the percentage uncertainty in the experimental value of g?