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Chapter 1 - Units, measurement and uncertainty

Classical Physics

The division between classical physics and modern physics is largely historical, classical physics is used where quantum effects and relativity can be neglected, and was largely developed before the beginning of the 20th century. This course will cover all this knowledge in two semesters (so we'll be moving fast!).

PHY131 PHY132
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Oscillations    Light


Measurements are made with respect to a standard which defines a unit.

  • How far/big? [m]
  • How much? [kg]
  • How much time [s]

These three quantities are the base SI units for mechanics, all other units in mechanics are derived. This makes dimensional analysis a useful test of an equation, especially when calculus is taken in to account.

There are many non-SI units in common usage (especially in the USA!) . You need to know how to convert from these units to SI units and vice-versa, though in practice we won't do too much of that in this class.

Unit prefixes

More on Wikipedia and in text.

Prefix  Abbreviation  Value
femto $f$ $10^{-15}$
pico $p$ $10^{-12}$
nano $n$ $10^{-9}$
micro $\mu$ $10^{-6}$
milli $m$ $10^{-3}$
centi $c$ $10^{-2}$
deci $d$ $10^{-1}$
kilo $k$ $10^{3}$
mega $M$ $10^{6}$
giga $G$ $10^{9}$
tera $T$ $10^{12}$

Scientific Notation

Large and small numbers are best written in scientific notation.


$0.000056\,\mathrm{m} = 5.6 \times 10^{-5}\,\mathrm{m\,or\,}5.6 \times 10^{-2}\,\mathrm{mm}$.

$795,000\,\mathrm{g} = 7.95 \times 10^{5}\,\mathrm{g\,or\,} 7.95 \times 10^{2}\,\mathrm{kg\,or\,}795\,\mathrm{kg}.$

In general it is not correct to give more significant figures for a number than the precision to which you know it. However you should not round off numbers too early in a calculation, as this can affect the accuracy of the final answer.

Accuracy and precision

While the words accuracy and precision sound like they refer to similar things their meanings in physics are actually slightly different.

Precision refers to how well a quantity can be determined. This determination of this quantity may be the result of multiple independent measurements, which presumably would improve the precision, but when we talk of precision we are not considering how the value of the quantity compares to a “known” or “established” value.

Accuracy, on the other hand, does make this comparison. The accuracy of a measurement refers to how well a measured value agrees with a a “known” or “established” value.


  • Precision - I measure the length of an object with a ruler and I am confident that my measured value, in meters, is correct to 3 decimal places. I can now say that my measurement has a precision of 1mm.
  • Accuracy - The object I measured is actually a standard length whose length is known absolutely (because it's a standard!), the difference between my measured value and the known correct value is the accuracy of my measurement.

Visual comparison of the difference between precision and accuracy

Let's play some darts and characterize various attempts to hit the target.

Error and Uncertainty

The precision to which we can measure something is limited by experimental factors, leading to uncertainty.

The deviation of a measurement from the “correct” value is termed the error, so error is a measurement of how inaccurate our results are. There are two general types of errors.

  • Systematic Errors - A error that is constant from one measurement to another, for example, an incorrectly marked ruler would always make the same mistake measuring something as either bigger or smaller than it actually is every time. These errors can be quite difficult to eliminate!
  • Random Errors - Random errors in your measurement occur statistically, ie. they deviate from the correct value in both directions. These can be reduced by repeated measurement.

But here is where it gets confusing… When you estimate the uncertainty of your measurement, as you will do frequently in the lab component of this course, you should consider the possible sources of error that contribute to the uncertainty. This way if there are large sources of error in your experiment, you will have a large uncertainty which will not exclude the accurate value of the quantity you are trying to measure.

Stating uncertainty

Because we cannot know any experimentally measured value to absolute certainty we state such values in the format


This is equivalent to saying that the actual value of the quantity could be anywhere between the value minus the uncertainty and the value plus the uncertainty.

Much of our activity in the lab will be about the best way to realistically determine and estimate for the uncertainty in any value we obtain.

Example of systematic error

An easily accessible standard is the offical US time kept by NIST. This can be compared to a watch to evaluate the systematic error in the time we should consider if we were to use it to time an experiment. This would probably only concern us if we needed to know the actual time that an event occurred at, rather than the difference in time for two events measured with the same watch.

Example of random error

Suppose in the video below I wanted to know the number of cars in the ring, while sitting in the van at the bottom of the screen. One way I could do try this is to measure the distance between two cars, and use this (presuming I know what the circumference of the ring is) to estimate the number of cars in the ring. Of course as we can see that depending on when I made the measurement I would either underestimate or overestimate the number of cars in the ring, we can see how the changing density of the cars both with position and time leads to a random error.

Averaging and Standard Deviation

When a leading cause of uncertainty in our measurement is random error we can lower the uncertainty in our measurement by repeated measurement and averaging (if the main sources of error are random). If we can assume that our measurements are governed by a typical statistical distribution then the standard deviation becomes a useful measurement of the variance of our data.

Average of N measurements: $\overline{t}=\frac {\sum t_{i}}{N} $

Deviation, or, how much does an individual measurement differ from the mean value: $t_{i}-\overline{t}$

The standard deviation of a measurement is given by: $\Delta t=\sqrt{\frac {\sum (t_{i}-\overline{t})^2}{(N-1)}}$

The standard deviation of the mean, or Standard Error is given by: $\Delta {\overline{t}}=\sqrt{\frac {\sum (t_{i}-\overline{t})^2}{N(N-1)}}$

The standard deviation of the mean gives an estimate of how much the calculated mean value may differ from the true mean value.

Propagation of uncertainty

Very often we need to make more than one measurement, or manipulate a measured quantity in an equation to find the quantity we really want. In these cases we need to propagate uncertainty. If the quantity we want to know is $f(x)$ and there is some variation in the measured quantity $x$ we can deduce the variation in $f(x)$ using calculus.

$\delta f = \frac{df}{dx}\delta x$

We should first square this (as the sign of the variation is not important to us).

$\delta f^{2} = (\frac{df}{dx})^2\delta x^{2}$

and then take the average of our mutliple variations

$\langle \delta f^{2} \rangle = (\frac{df}{dx})^2\langle \delta x^{2} \rangle$

Taking the square root gives us the relationship we are looking for

$\sqrt{\langle \delta f^{2} \rangle} = |\frac{df}{dx}|\sqrt{\langle \delta x^{2} \rangle} \rightarrow \Delta f = |\frac{df}{dx}|\Delta x$

Propagation with two variables

If $f(x,y)$ then $\delta f = \frac{\partial f}{\partial x}\delta x + \frac{\partial f}{\partial y}\delta y$

Now $\delta f^{2} = (\frac{\partial f}{\partial x})^2\delta x^2 + (\frac{\partial f}{\partial y})^2\delta y^2 + \frac{\partial f}{\partial x}\frac{\partial f}{\partial y}\delta x \delta y$

And $\langle \delta f^{2} \rangle = (\frac{\partial f}{\partial x})^2\langle\delta x^2 \rangle + (\frac{\partial f}{\partial y})^2\langle\delta y^2 \rangle + \frac{\partial f}{\partial x}\frac{\partial f}{\partial y}\langle\delta x \delta y\rangle$

Does the variation in $x$ affect the variation in $y$? If not we can say that these variables are uncorrelated and that $\langle\delta x \delta y\rangle = 0$, which leads us to

$\Delta f = \sqrt{(\frac{\partial f}{\partial x})^2\Delta x^2 + (\frac{\partial f}{\partial y})^2\Delta y^2}$

Some propagation rules

From $\Delta f = |\frac{df}{dx}|\Delta x$ and $\Delta f = \sqrt{(\frac{\partial f}{\partial x})^2\Delta x^2 + (\frac{\partial f}{\partial y})^2\Delta y^2}$

we can derive some useful rules for common operations you will carry out in the labs. You can find more details about these equations in the error manual

if $S=aX$ then $\Delta S=a\Delta X$

if $S=A\pm B$ then $\Delta S=\sqrt{(\Delta A)^2+(\Delta B)^2}$

if $S=A\times B$ or $\frac{A}{B}$ then $\large \frac{\Delta S}{S}=\sqrt{(\frac{\Delta A}{A})^2+(\frac{\Delta B}{B})^2}$

if $S=A^n$ then $\large \frac{\Delta S}{S}=|n|\times \frac{\Delta A}{A}$

Measurement Exercises

Using the meter stick

1. Determine the length of the perimeter of one of the carpet tiles on the floor. Estimate the uncertainty in your value by considering the uncertainty in your measurement of each side.

2. Determine the area of one of the carpet tiles on the floor. Estimate the uncertainty in your value by considering the uncertainty in your measurement of each side.

3. Determine the area of one of the round tables you are sitting at. Estimate the uncertainty by considering the uncertainty in your measurement of the diameter of the table.

Finally, let's collect the values from the different groups and average them and then estimate the uncertainty in our average.

Why is error and uncertainty so important?

An accurate estimate of the uncertainty in an experiment is the only way to determine whether an experiment is consistent or inconsistent with a theory.

If the theoretical prediction lies within the estimate of the uncertainty of the experiment then we can say the theory is consistent with the experiment. Another way of stating this would be that the measured value is consistent with the theoretical one if the error is less than the uncertainty. However, if the uncertainty is very large this this may be a meaningless statement! If we estimate the uncertainty to be smaller than it really is we may discard a valid theory (and perhaps an important discovery).

Our aim is therefore always to accurately estimate the uncertainty of our results and strive to improve it!

phy131studiof16/lectures/chapter1.txt · Last modified: 2016/08/29 08:29 by mdawber
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