Chapter 01 : Introduction, Measurement, ...

 

 

Units, Standards, and the S.I. system

 

Two 'metric' systems

  • CGS system : centimeters, grams, seconds (chemistry, biology, ...)

  • MKS system : meters, kilograms, seconds (pretty much everybody else)

 

 

How are these units defined?

 

  • Previously: physical - a 'standard kilogram', etc

  • Currently: define in terms of universal constants

  • Speed of light now defined to be exactly 299,792,458 m/s

  • The second is now defined in terms of a particular oscillation (the 'hyperfine transition frequency') of cesium-133 : to be exactly 9,192,631,770 cycles in one second

  • and so on...

 

 

Measurement Uncertainty, Significant Figures

 

Length of a (nominally 100 cm long) stick, as measured by many different people (the chart here is the measurement minus 100 to highlight the differences):

 

Characterizing the results:

  • Midpoint and half-width: 104 ± 16 cm

  • Mean and standard deviation: 103.8 ± 4.3 cm

  • Expected (average) value +/- some uncertainty: x ± Δ x

  • Fractional uncertainty: Δx / x   ( 0.0414.. )

  • Percentage uncertainty: 100 ( Δx / x )   ( 4.14% )

Gaussian or 'normal' distribution

\[ f(x) = \frac{1}{\sqrt{2\pi \sigma^2 }}e^{-\frac{ (x-\mu)^2 }{2\sigma^2} } \]

 

 

Significant Figures

 

Nearly all measurements have some uncertainty associated with them.

Convention:

  • x=1.2 implies uncertainty of ± 0.05

  • x=1.20 implies uncertainty of ± 0.005

  • x=1.200 implies uncertainty of ± 0.0005

 

What about a number like 1200? Are those zeros meaningful or just padding?

 

Scientific notation

 

  • 1.2 × 103 implies ( 1.2 ± 0.05 ) × 103 or 1200 ± 50

  • 1.20 × 103 implies ( 1.20 ± 0.005 ) × 103 or 1200 ± 5

 

 

Why do we care? Propagation of Errors

 

If all measurements have at least some error associated with them:

  • How do these uncertainties affect calculations?

  • If (say) we need a result that has (no more than) a 1% error, how accurately do we need to measure the inputs to the calculation?

 

The width and length of a rectangular area are measured multiple times, yielding the results shown in the figure.

What is the area of this rectangle (including uncertainty)?

What effect do the 'input' uncertainties have on the result?

Volume of a Sphere

We estimate the diameter of a spherical globe to be:

\[ D = (40 \pm 2)~cm \]

What is the volume (including the uncertainty)?

If we need the volume more accurately (say a 1% uncertainty) how accurately do we need to determine the diameter?

 

 

 

Worked-out Examples

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