Introduction and Survey

Appendices

Summary

• Physics deals with the basic building blocks of our world, their mutual interactions and the synthesis of material from these basic particles.

• The gain of knowledge is pushed by specific experiments. Their results serve for the development of a general theory of nature and to confirm or contradict existing theories.

• Experimental physics started in the 16th century (e. g. Galilei, Kepler) and led to a more and more refined and extensive theory, which is, however, even today not yet complete and consistent.

• All physical quantities can be reduced to three basic quantities of length, time and mass with the basic units 1 m, 1 s, and 1 kg. For practical reasons four more basic quantities are introduced for molar mass (1 mol), temperature (1 K), electric current (1 A) and the luminous power (1 cd).

• The system of units which uses these basic $3+4$ units is called SI-system with the units 1 m, 1 s, 1 kg, 1 mol, 1 K, 1 A and 1 cd.

• Every measurement means the comparison of the measured quantity with a normal (standard).

• The length standard is the distance which light travels in vacuum within a time interval of $(1/299{,}792{,}458)$ s. The time standard is the transition frequency between two hyperfine levels in the Cs atom measured with the caesium atomic clock. The present mass standard is the mass of the platinum-iridium kilogram, kept in Paris.

• Each measurement has uncertainties. One distinguishes between systematic errors and statistical errors. The mean value of n independent measurements with measured values x _i is chosen as the arithmetic mean

  $$\displaystyle\overline{*}{x}=\frac{1}{n}\sum^{n}_{i=1}x_{i}{\;},$$

  which meets the minimum condition

  $$\displaystyle\sum^{n}_{i=1}(\overline{*}{x}-x_{i})^{2}=\text{minimum}.$$

  If all systematic errors could be eliminated the distribution of the measured values x show the statistical Gaussian distribution

  $$\displaystyle f(x)\propto{\text{e}}^{-(x-x_{\text{w}})^{2}/2\sigma^{2}}{\;},$$

  about the most probable value, which equals the true value $x_{\mathrm{w}}$. The half-with of the distribution between the points $f(x_{\mathrm{w}})/e=f(x_{\mathrm{w}}\pm\sigma)$ is $\sigma\cdot\sqrt{2}$ Within the range $x=x_{\mathrm{w}}\pm\sigma$ fall 68% of all measured values. The standard deviation σ of individual measurements is

  $$\displaystyle\sigma=\sqrt{\frac{\sum(\overline{*}{x}-x_{i})^{2}}{n-1}}{\;},$$

  the standard deviation of the arithmetic means is

  $$\displaystyle\sigma_{\text{m}}=\sqrt{\frac{\sum(\overline{*}{x}-x_{\text{i}})^{2}}{n(n-1)}}{\;}.$$

  The true value $x_{\mathrm{w}}$ lies with the probability of 68% within the interval $x_{\mathrm{w}}\pm\sigma$, with a probability of 99.7% in the interval $x_{\mathrm{w}}\pm 3\sigma$. The Gaussian probability distribution for the measured values x _oi has a full width at half maximum of

  $$\displaystyle\Delta x_{1/2}=2\sigma\sqrt{2\cdot\ln 2}=2.35\sigma{\;}.$$

Problems

1.1

The speed limit on a motorway is 120 km/h. An international commission decides to make a new definition of the hour, such that the period of the earth rotation about its axis is only 16 h. What should be the new speed limit, if the same safety considerations are valid?

1.2

Assume that exact measurements had found that the diameter of the earth decreases slowly. How sure can we be, that this is not just an increase of the length of the meter standard?

1.3

Discuss the following statement: „The main demand for a length standard is that its length fluctuations are smaller than length changes of the distances to be measured“.

1.4

Assume that the duration of the mean solar day increases by 10 ms in 100 years due to the deceleration of the earth rotation. a) After which time would the day length be 30 hours? b) How often would it be necessary to add a leap second in order to maintain synchronization with the atomic clock time?

1.5

The distance to the next star (α-Centauri) is $d=4.3\cdot 10^{16}\,\mathrm{m}$. How long is the travelling time of a light pulse from this star to earth? Under which angle appears the distance earth-sun from α-Centauri? If the accuracy of angular measurements is $0.1^{\prime\prime}$ what is the uncertainty of the distance measurement?

1.6

A length L is seen from a point P which is 1 km (perpendicular to L) away from the centre of L, under an angle of $\alpha=1^{\circ}$. How accurate can the length be determined by angle measurements from P if the uncertainty of α is $1^{\prime}$?

1.7

Why does the deviation of the earth orbit from a circle cause a variation of the solar day during the year? Give some arguments why the length of the mean solar day can change for different years?

1.8

How many hydrogen atoms are included in 1 kg of hydrogen gas?

1.9

How many water molecules H${}_{2}$O are included in 1 litre water?

1.10

The radius of a uranium nucleus (A = 238) is $8.68\cdot 10^{-15}\,\mathrm{m}$. What is its mean mass density?

1.11

The fall time of a steel ball over a distance of 1 m is measured 40 times, with an uncertainty of 0.1 s for each measurement. What is the accuracy of the arithmetic mean?

1.12

For which values of x has the error distribution function $\exp[-x^{2}/2]$ fall to 0.5 and to 0.1 of its maximum value?

1.13

Assume the quantity x = 1000 has been measured with a relative uncertainty of $10^{-3}$ and y = 30 with $3\cdot 10^{-3}$. What is the error of the quantity $A=(x-y^{2})$?

1.14

What is the maximum relative error of a good quartz clock with a relative error of $10^{-9}$ after 1 year? Compare this with an atomic clock ($\Delta\nu/\nu=10^{-14}$).

1.15

Determine the coefficients a and b of the straight line $y=ax+b$ which gives the minimum squared deviations for the points $(x,y)=(0,2)$; $(1,3)$; $(2,3)$; $(4,5)$ and $(5,5)$. How large is the standard deviation of a and b?