Series 2, Year 19

In tram at rest the pen is suspended on a sting of length $l$. The mass of the pen is $m$. The tram accelerates with constant acceleration $a$. Calculate the maximum angle of displacement of the pendulum (maximum angle between the string and the vertical direction) and the time when it crosses the starting position.

An engine with 8 carriages each of mass 40 t is accelerating to maximum speed of 120 km ⁄ h at the rails long 1 km. What must be the minimum weight of the engine to accelerate without wheel spinning over the rails?

Assume the static friction coefficient $f=0,2$. Air friction is negligible.

A emission spectral line of Helium was observed in spectrum of a star. The wavelength of helium line is 587,563 nm . However, the observed line in spectroscope was blurred between 587,60 nm and 587,67 nm . Estimate the temperature of the star and its speed in the space. How is this blurring caused?

Derive the temperature dependence of heat conductivity of metals, if the temperature dependence of electrical conductivity is known.

For free electrons in metal the model of ideal gas can be used, e.g. free electrons are moving without external forces on straight lines (ions are not considered) and sometimes collides with other electrons, when they change the direction and amplitude of velocity.

The heat transferred by crystal lattice is negligible to the heat transferred through free electrons. Each electron has heat capacity $c$, which is temperature independent.

Explain, why is it possible to 'over-blow' (play one octave higher than normally) a flute, and it is not possible with clarinet.

Measure pressure of gas in pressurised CO_{2} container used for home making of sparkling lemonade from still water. The container is filled with CO_{2} or is filled with N_{2}O for fluffy cream preparation.

• What is the relation between the number of microstates Ω($E)$ of thermostat with energy ≤ $E$ and quantity defined as $η(E)$ (e.g. the number of microstates with energy in interval $E±Δ)$ for small Δ?
• Assume a system of $N$ independent harmonic oscillators, where energy of each oscillator can be one of following values:$nhω$, where $n=0,1,2$,…

(neglecting energy of zero oscillations). What will be the quantity $η(E)$ and $β(E)$ for big $N$ and $E?$

• Find the same quantity as in the previous case for system of $N$ non-interacting free electrons trapped on line or in square or in cube, respectively.

Hint: Use de Broglie relation between momentum and wavelength of de Broglie wave. The integer multiple of half-wavelengths must fit on the line. De Broglie wavelength in square can be imagined as product of waves in direction of axis $x$ and $y$, quantum condition is similar to the condition for line.