Assignment of Series 2 of Year 29

A rat is running on ice with speed $v$. Suddenly he decides to turn 90$°$ so that he keeps running with the same speed in the new direction. What is the least amount of time he needs for such a turn? Suppose that rat's feet can move independently. Coefficient of friction between rat's feet and ice is $f$.

Once in a while, a situation may occur, that the nominal value of coins is lower that their manufacturing costs. Assume we have two coins, made of a gold-silver alloy. The first one has diameter $d_{1}=1\;\mathrm{cm}$, second one $d_{2}=2\;\mathrm{cm}$, both have thickness $h=2\;\mathrm{mm}$. If we submerge them in mercury, the smaller one sinks to the bottom, whilst the larger one rises to the surface. If we submerge both coins, smaller one on top of the larger, they neither rise nor sink. Assuming the smaller coin is made of pure gold, determine the fraction of silver in the larger coin (in percent of mass).

Bonus: How would the result change if the smaller one could contain silver as well?

From a spaceship on a circular orbit with height $h=2000\;\mathrm{km}$ above the surface of Earth a screwdriver is thrown with speed $v=5\;\mathrm{km}\cdot h^{−1}$ relative to the rocket towards the center of the Earth. Determine when will the screwdriver hit the surface?

Consider an optical system composed of three semitransparent mirrors placed behind each other along one axis. Every mirror by itself reflects half of incident light and lets the other half pass. Determine what fraction of light passes through the system of mirrors.

Bonus: Solve the problem for $n$ such mirrors.

Mirek felt that during winter it is a little bit too dark for reading in his room. So he decided to make a hole in his wall for another window. He went to glass-works first to buy the glass panes. There was one nice round piece, but before he would buy it, he needed to check whether it's not too uneven (specifically convex). He placed the pane on the glass desk of the glass-works and saw rainbow circles around the centre of the pane caused by interference of the perpendicularly coming white light in the thin space between the two glasses. Mirek randomly chose two red circles ($λ≈700nm)$ and measured with a ruler their diameters $d_{k}=(10,5±0,5)\;\mathrm{mm}$ and $d_{k+1}=(13,0±0,5)\;\mathrm{mm}$. Based on these measurements he managed to determine the radius of curvature of the pane. Calculate it as well and think about it's errors.

Imagine that an intelligent seven-year-old turns to you with a question: „What exactly is this superconductivity?“ What would you have to explain and teach him first, in order to reasonably clarify this phenomenon without using „lies-to-children“ (the term was first described in The Science of Discworld novel; it describes an explanation, that helps explaining a complex subject by simplifying elementary explanations, so that they are understandable, though technically wrong, e.g. imaging atoms as tiny solid balls). Try to elaborate the answer as much as possible.

Buy any effervescent (i.e. fizzy) tablets and measure the time that takes for the tablet to fully dissolve in water as a function of temperature of this water. Discuss the possible causes and propose why is the relation the way it is.

• Which types of processes (isobaric, isochoric, isothermal and adiabatic) can be reversible?
• Take the relation

$T=\frac{pV}{nR}\$,,

where $n=1mol$, $p=100kPa$ and $V=22l$. How will $T$ change, if we change both $p$ and $V$ by 10$%$, by 1$%$ or by 0$,1%?$ Calculate it in two ways: precisely and by using the relation: $$\;\mathrm{d} T=T_{,p} \mathrm{d} p T_{,V} \mathrm{d} V .$$

What is the difference between the results?

• d gymnastics:
• Show that

$$\;\mathrm{d} (C f(x)) = C \mathrm{d} f(x)\,,$$

where $C$ is constant.

• Calculate

$$\;\mathrm{d} (x^2) \ \quad \mathrm{a} \quad \mathrm{d} (x^3).$$

• Show that

$$\;\mathrm{d}\left( \frac 1x \right)= -\frac {\mathrm{d} x}{x^2}$$

from the definition, that is $$\;\mathrm{d} \left(\frac 1x \right)= \frac {1}{x \mathrm{d} x} - \frac 1x$$

This might be handy: $(x \;\mathrm{d} x)(x-\mathrm{d}$ x) = x^2 - (\mathrm{d} x)^2 = x^2$\$,.

**Bonus:** $This$ holds 

$$\sin \;\mathrm{d} \vartheta = \mathrm{d} \vartheta \quad a \quad \cos \mathrm{d} \vartheta = 1.$$

And you have the addition formula as well $$\sin (\alpha \beta ) = \sin \alpha \cos \beta \cos \alpha \sin \beta,$$

Prove $$\;\mathrm{d}\left( \sin \vartheta \right)=\, \mathrm{d} \vartheta \cos \vartheta .$$

**Bonus:** Similarly show 

$$\;\mathrm{d} \left(\ln x \right)= \frac{\mathrm{d}x}{x}$$

using $$\ln (1 \;\mathrm{d} x) = \mathrm{d} x$$

• Explain, why isobaric temperature is lower than isochoric.