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An Introductory Physics Course with Peter Eyland
Lecture 2 (Measurement)

In this lecture the following are introduced:
•Laboratory instruments: the balance, Vernier calipers, Micrometers, stopwatches, measuring cylinders,
•Uncertainty and accuracy: random and systematic errors,
•Accumulating errors in the laboratory,
•Graphing with errors,
•Volume and area formulae.

 

Laboratory instruments

The balance
There are balances available in the Laboratory that you can use to measure the mass of an object up to a few kilograms.
The balance should be "zeroed" before use by using the screw adjustments (this may take some time so if it's OK don't change anything).
If a liquid is to be weighed then the mass of the container must be subtracted from the total.
The accuracy of the balance should be estimated and this should be reflected in the number of significant digits.

 

Vernier Calipers
Vernier calipers are available in the Laboratory which you can use to measure lengths up to 200mm.
A vernier scale enables you to read to 1/10 of the smallest main scale division of 1mm, i.e. 0.1mm. Note that if the object is irregularly shaped the accuracy of a measurement may have to be determined by making measurements at a number of places.
Systematic errors may arise if the calipers are hotter or colder than its reference temperature. If the scale has expanded due to heating then the apparent measurements will be always be read as smaller than they really are.
Likewise if the scale has contracted because of the cold then the apparent measurements will always be read as larger.

 

Micrometers
Micrometers are generally available in the Laboratory which enable you to read to 1/100 of the smallest main scale division of 1mm, i.e. 0.01mm.
Each complete turn of the barrel moves the head by 0.5mm. The barrel scale has 50 divisions to divide the 0.5mm into 0.01mms.
The "zero" has to be established by using the ratchet to close the heads.
Again, the accuracy of the meter may exceed variations in the object.

 

Stopwatches
Stopwatches are used to measure time intervals.
There may be systematic errors if the clock runs fast or slow.
There will be random errors from estimating when to stop the clock and also from your reflexes that will slow as the day progresses.
This means that the error will usually be more than the smallest time measurement of 0.001s.

 

Measuring cylinders
Measuring cylinders are used to measure the volume of a liquid.
Different sizes enable you to measure a range of volumes.

 

Uncertainty and accuracy

The carpenter's rule is "measure twice and cut once", so in Physics we rely on taking a number of measurements to improve our confidence in the result. A number of measurements gives both a mean (or average) for the measurement and a variation for the uncertainty (or error).
A "mistake" is a human error that occurs by misreading a scale, or neglecting a zero error, etc.
An "error" is an estimate of measurement accuracy that is stated as an uncertainty.
Human errors (in the form of mistakes) are not accepted in Laboratory measurements but measurement error estimates are necessary.
The mean is calculated from a number of measurements by finding the sum and dividing by the number of measurements.
An estimate of the error is given by the largest difference from the mean, though this rule often has to be modified because it overestimates the error.

Random errors occur because of measurement technique, reaction time variations, variations in the system conditions between measurements (temperature, pressure etc), or simply variations in object dimensions. They have no pattern and cannot be completely eliminated.

Systematic errors always push the measurement higher or lower. They occur because the scale on the instrument may have changed with temperature, or some physical element within the measuring instrument (e.g. a spring) may have changed due to age or humidity etc. The latter are called "calibration" errors.

An overall uncertainty has to be calculated when a quantity is calculated by addition, subtraction, multiplication or division of measurements which have error estimates. The textbook does this is by working out the extreme cases.

Example:
A rectangular sheet of metal has
length measurements, 252mm, 247mm, 251mm, 247mm, 249mm ,and
breadth measurements, 100mm, 102mm, 99mm, 98mm, 101mm.
Find the area and its error.

equation

There are 3 significant figures in the measurements so the mean is written as 249mm.

The largest difference from the mean is 3mm.
The result is then 249±mm, but this seems a bit too much because only one of the five measurements is 3mm from the mean.
Four out of the five are within 2mm.
It is then better to estimate it as 249±2mm.

equation
This has the correct number of significant figures, and the largest variation is 2.
The result is 100±2mm

Two out of the five are within 2mm.
Three out of the five are within 1mm.
It is better to estimate it as 100±1mm.

The largest area is 251 x 101 = 25351mm2
The smallest area is 247 x 99 = 24453mm2
The final result is 24900±400mm2

 

Accumulating errors in the Laboratory

The laboratory manual (not included here) gives another way of accumulating errors, which you should use.
The absolute error of a quantity is simply the variation in that quantity [equation].
The relative error of a quantity is the fraction (or percentage) of the variation to that quantity [ or equation].

For errors that are based on the same source of error, the following applies:
• When quantities are added or subtracted, the absolute errors are added.
• When quantities are multiplied or divided, the relative errors are added.

Example
The mass of an object was found to be 9.0±0.1g, and its volume from the difference between 98.00±0.05ml and 97.00±0.05ml.
Find the density.

The volume is given by the difference in the volume measurements: 98.00 - 97.00 = 1.00ml
The error in the volume is the sum of the absolute errors: 0.05±0.05 = 0.1ml.

The volume with its error is written as: 1.0±0.1ml

The density is given by the quotient: mass per unit volume.

equation

The mass has a relative error of equation.

The volume has a relative error of equation

The relative error in the density (a quotient) is the sum of the relative errors, i.e.

=10% +1.1%= 11%

The absolute error in the density is then equation
The density and error is equation.

The number ranges from 8010 to 9990, but since the first digit changes, it is probably best to write the density as .

Graphing

In practice, all experimentally determined data are subject to the uncertainties in accuracy called "errors". When experimental data are presented in a graphical format, the measurements are drawn with "error bars" which give the range of uncertainty.

For example
graph

In this case, there appears to be a linear (or straight line) relationship between the quantities.

Three slopes can be found. The two "worst lines of best fit", and then the line of best fit.

The steepest and shallowest slopes that go through all the error bars are first drawn.
graph

Then using the point where the two worst lines cross, the best-fit line is drawn.
graph

The best-fit line gives the mean slope and the error in the slope comes from (usually) the largest difference from the mean. If there is no line that can be drawn through all the error bars, then either the errors have been underestimated, or there is no straight line relationship. Most experimental relationships can be plotted to give a straight line (with sufficient ingenuity!) and this can give the constants of the relationship.

Example
Using a graphical technique, it is possible to determine the stiffness factor of a spring by hanging it vertically, suspending a mass from its lower end, setting the mass into vertical (up and down) oscillation, and measuring the period of that vibration.

Theory gives that for a spring of stiffness factor, k, and mass, m, if a mass, M, is suspended from it and set into vertical oscillation, the period of oscillation, T, is given by:

equation

For each of a number of different masses suspended from the spring, the period of oscillation of the mass is measured.

To produce a straight-line graph, the relationship has to be squared and re-arranged:

equation

A plot of M against T2 should yield a straight line graph.
• the slope of the graph would be equation
• the intercept on the vertical axis would be equation.
This would need to be projected in any graph because a zero mass is not allowed.

The following table, gives experimental data with estimates of the uncertainties.

M

ΔM

T

ΔT

equation

T2

equation

ΔT2

kg

kg

s

s

-

s2

-

s2

0.0500

0.0003

0.48

0.05

10

0.23

20

0.05

0.0700

0.0004

0.56

0.05

9

0.31

18

0.06

0.1000

0.0006

0.63

0.05

8

0.40

16

0.06

0.1500

0.0009

0.67

0.05

7

0.45

14

0.06

0.2000

0.0010

0.77

0.05

6

0.59

12

0.07

0.2500

0.0015

0.89

0.05

6

0.79

12

0.09

0.3000

0.0020

0.93

0.05

5

0.86

10

0.09

0.3500

0.0020

0.98

0.05

5

0.96

10

0.10

0.3700

0.0020

1.03

0.05

5

1.06

10

0.11

graph The errors in the mass are too small to worry about, but the errors in the time are significant.
Drawing the three slopes (note that "joining the dots" doesn't give any useful information).
graph
The three slopes are 0.46, 0.41, and 0.35.
This means the slope and error is 0.41±0.05 kg.s-2.

The spring constant is
equation

The error in the spring constant k, will have the same %error as the slope because the multipliers are pure numbers with no error.

The slope has a equation error.
The error in spring constant k, is equation
This shows that the "units" column is in doubt, so the answer is 162 N.m-1.
The error in the time has dominated.

Area and volume formulae

equation

________________________________

equation

________________________________

equation

________________________________

equation

 

Summarising:

In the Laboratory the following instruments are used:
mass balances, Vernier calipers, Micrometers, stopwatches and measuring cylinders.
Random errors occur because of measurement technique, reaction time variations, variations in the system conditions between measurements (temperature, pressure etc), or simply variations in object dimensions.
Systematic errors always push the measurement higher or lower. They occur when the scale on the instrument has changed with temperature, or some physical element within the measuring instrument (e.g. a spring) has changed due to age or humidity etc.
When adding or subtracting quantities you add the absolute errors.
When multipying or dividing quantities you add the relative errors.
When graphing with error bars: the best-fit line gives the mean slope and the error in the slope comes from (usually) the largest difference from the worst-fit lines.
Volume and area formulae are assumed knowledge.

 


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