1.2 Measurements & Errors

A student uses a digital ammeter to measure a current. The reading of the ammeter is found to fluctuate between 1.98 A and 2.02 A.

The manufacturer of the ammeter states that any reading has a systematic uncertainty of ±1%. 

Which value of current should be quoted by the student?

The uncertainty in the value of the momentum of a trolley passing between two points X and Y varies with the choice of measuring devices.

 Measurements for the same trolley made by different instruments were recorded. 

 Which measurements, one for each quantity measured, leads to the least uncertainty in the value of the momentum of the trolley?

In an experiment to determine the acceleration of free fall g, a ball bearing is held by an electromagnet. When the current to the electromagnet is switched off, a clock starts and the ball bearing falls. After falling a distance h, the ball bearing strikes a switch to stop the clock which measures the time t of the fall.

If systematic errors cause t and h to be measured incorrectly, which error must cause g to appear greater than 9.81 m s^–2?

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 115 ± 1 mm on the adjacent rule and the lower mark at 385 ± 1 mm. The ball passes the top mark at 1.50 ± 0.02 s and passes the lower mark at 3.50 ± 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?

A fixed quantity x₀ is measured many times in an experiment that has experimental uncertainty. A graph is plotted to show the number n of times that a particular value x is obtained.

 Which graph could be obtained if the measurement of x₀ has a large systematic error but a small random error?

A micrometer screw gauge is used to measure the diameter of a copper wire.

 The reading with the wire in position is shown in diagram 1. The wire is removed and the jaws of the micrometer are closed. The new reading is shown in diagram 2.

What is the diameter of the wire?

A student finds the density of a liquid by measuring its mass and volume. The following is a summary of his measurements.

       mass of empty beaker = (20 ± 1) g

      mass of beaker + liquid = (70 ± 1) g

      volume of liquid = (10.0 ± 0.5) cm³

 He correctly calculates the density of the liquid as 5.0 g cm^–3.

 What is the uncertainty in this value?

A steel wire is stretched in an experiment to determine the Young modulus for steel.

The uncertainties in the measurements are given below. 

What is the percentage uncertainty in the Young modulus?

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 relates to the experiment?

In an experiment, a radio-controlled car takes 2.50 ± 0.05 s to travel 40.0 ± 0.1 m.

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

Four students each made a series of measurements of the acceleration of free fall g. The table shows the results obtained.

 Which set of results could be described as precise but not accurate? 

Which experimental technique reduces the systematic error of the quantity being investigated?

A steel rule can be read to the nearest millimetre. It is used to measure the length of a bar whose true length is 895 mm. Repeated measurements give the following readings. 

Are the readings accurate and precise to within 1 mm? 

The Young modulus of the material of a wire is to be found. The Young modulus E is given by the equation below.

The wire is extended by a known force and the following measurements are made.

 Which measurement has the largest effect on the uncertainty in the value of the calculated Young modulus? 

The angular deflection of the needle of an ammeter varies with the current in the ammeter as shown in the graph.

Which diagram could represent the appearance of the scale on this meter?

Four different students use a ruler to measure the length of a 15.0 cm pencil. Their measurements are recorded on four different charts.

Which chart shows measurements that are precise but not accurate?

In the circuit shown, an analogue ammeter is to be recalibrated as a thermometer. The graph shows how the resistance R of the thermistor changes with temperature T.

Which diagram could represent the temperature scale on the ammeter?

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.

The spring constant is 100 ± 2 N m^–1 and the extension is 0.050 ± 0.002 m.

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