Problem Statement of Series 3, Year 29

In an aquarium of a spherical shape with the radius of $r=10\mathrm{cm}$ which is completely filled with water, swim two identical fish in opposite directions. The fish has a cross-sectional area of $S=5\mathrm{cm}$, Newton's drag coefficient $C=0.2$ and it swims with a speed of $v=5\mathrm{km}\cdot h^{-1}$ relative to the water. How long have the fish to swim in the aquarium to increase the temperature of the water by 1 centigrade?

The young alchemist George has learnt to measure electrochemical equivalents. He measured quite precisely the electrochemical equivalent $A=(6.74\pm 0.01)\cdot {10}^{-7}\mathrm{kg}\cdot C^{-1}$ of an unknown sample. How can he determine what substance was his sample made of?

Consider a massless spring with spring constant $k$. Weights are attached to both ends with masses $m$, and $Mrespectively$. This system is placed on a horizontal surface so that weight of mass $Mlies$ on the surface and the spring with the second weight points up. The system is in&nbspequilibrium (i.e. top weight does not oscillate) and length of the spring in this state is $l$. How much do we have to compress the spring so that the weight of mass $M$ jumps up when it is released? Consider only vertical motion.

After we press the break pedal, the car does not start to break immediately. During the time $t_r$ the breaking force grows linearly up to the maximum force $F_m$. Coefficient of static friction between the tire and a road is $f$. What is the maximum speed of car so that the car does not skid even during emergency breaking?

The notebook of a size of A4 (297 x 210 mm) lies on a desk with an inclination of $\alpha =5°$. The notebook weights $m$, between the desk and the notebook there acts a static friction force with coefficient $f_0=0.52$. Then, we hit the desk so it starts to oscillate (in the direction of the inclination of the desk) with a frequency $\nu =10\mathrm{Hz}$ and an amplitude $A=1\mathrm{mm}$.

• Determine by which extra force (perpendicular to the desk) we have to act on the notebook so it does not start to move.
• Determine how long it takes the notebook to fall off the desk if at the beggining its bottom edge (the shorter one) is at the bottom edge of the desk. Dynamic friction coeficient is $f$, consider notebook as a rigid plate.

Lukas has been weightlifting and he managed to make a black hole of mass 1 kg. As he isn't too fond of quantum field theory in curved spacetime, the black hole does not radiate. Lukas drops this hole and it begins oscillating within the earth. Try to estimate how long would it take for the mass of the black hole to double. Is it safe to make black holes at home?

Examine the dependence of a weight of a hydrogel ball on a time of submersion in a water and on a concentration of salt dissolved in water. Note We do not send the experimental material abroad, therefore the hydrogel you buy must be described in detail.

• All states of ideal gas can be shown on various diagrams: $pV$ diagram, $pT$ diagram and so on. The first quantity is shown is on vertical axis, the second on horizontal. Every point therefore determines 2 parameters. Sketch in a $pV$ diagram the 4 processes with ideal gas that you know. Do the same on a $Tp$ diagram. How would $UT$ diagram look like? Explain how would the unsuitability of these two variables appear on the diagram.

• What are the dimensions of entropy? What other quantities with the same dimensions do you know?

• In the text for this series we analysed a case of entropy increasing as heat flows into a gas. Perform a similar analysis for the case of heat flowing out of the gas.

• We know that entropy does not change during an adiabatic process. Therefore, the expression for entropy as a function of volume and pressure $S(p,V)$ can only contain a combination of pressure and volume that does not change during an adiabatic process.

What is this expression? Draw lines of constant entropy on a $pV$ diagram ($p$ on vertical axis, $V$ on horizontal). Does this agree with the expression for entropy we have derived?

• Express the entropy of an ideal gas as functions $S(p,V)$, $S(T,V)$ and $S(U,V)$.