Uncertainties & Errors (DP IB Physics: SL)

Systematic and random errors can be compared.

What are the properties that apply to random errors?

1  The error is consistently too high or too low and by different amounts

2  The error is constantly too high or low and by the same amount each time

3  The error can be fully eliminated

4  The error cannot be fully eliminated

5  The error can be reduced by averaging repeated measurements

6  The error cannot be reduced by averaging repeated measurements

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?

Analogue meters often have a mirrored or metallic strip behind the pointer. 

What is the main purpose of this strip? 

Two lengths, n and n – 2 have an uncertainty of . The uncertainty  is much less than n. 

In which of the following quantities is the percentage uncertainty the largest? 

In an experiment to determine the acceleration of free fall g, a student presents their calculated value as 9.7835 m s^–2 and states their value is only accurate to ± 5%.

Which of the following gives their calculated value to an appropriate number of significant figures? 

A small conker is attached to a string and whirled overhead in a horizontal circle. The tension in the string T is measured for a variety of different linear speeds v and the following graph is obtained.

These experimental results suggest that there is...

Which of the following experimental techniques does not reduce the random error associated with the quantity being measured? 

The length and breadth of a rectangular card was found to be (5.0 ± 0.2) cm and (3.0 ± 0.2) cm respectively. The area of the card cannot be more than: 

An object dropped from the edge of a cliff is timed to take (10.0 ± 0.1) s to fall to the surface of the sea. If the acceleration due to free fall is (10.0 ± 0.2) m s^–2, the calculated height of the cliff from which the object was released should be quoted as: 

A certain rule can be read to the nearest millimetre. It is used to measure the length of a piece of string whose true length is 50 mm. Repeated measurements of the string give the following readings:

Are the readings accurate and precise to within 1 mm? 

A researcher plots a graph to investigate the variation of current and potential difference of a resistor.

After realising they have read the manufacturers label of their electrical equipment incorrectly, they determine the size of the error bars shown should decrease.

Which row correctly describes how this will affect the value of the maximum gradient G_max, the minimum gradient G_min, and the resistance of the resistor R?

A mass is dropped from rest and falls through a distance of 2.0 m in a vacuum. An observer records the time taken for the mass to fall through this distance using a manually operated stopwatch and repeats the measurements a further two times.

The average result of these measured times, displayed in the table below, was used to determine a value for the acceleration of free fall. This was calculated to be 9.8 m s^–2.

Which statement best describes the quality of the measurements taken for this experiment?