Problem Statement of Series 2, Year 34

The length of daytime and nighttime varies during the year and it may vary differently in different places on Earth. What about the average length of daytime during one year? Is it the same everywhere or does it vary in different places? A qualitative description is sufficient.

Bonus Try to estimate the maximum difference between the average length of daytime and $12\mathrm{h}$.

Cathy and Catherine are watching a ship which is sailing with a constant speed towards a port. Cathy is standing on a rock above the port and her eyes are $h_1=20\mathrm{m}$ above the surface of the water. Catherine is standing under the rock and her eyes are $h_2=1.7\mathrm{m}$ above the surface of the water. If Catherine sees the top of the incoming ship $t=25\min$ after Cathy sees it, what is the time of arrival of the ship to the port? Assume that the Earth is a perfect sphere with a radius $r=6378\mathrm{km}$.

There are several movie scenes where a car falls into water together with its passengers. Calculate the torque with which a person must push the door in order to open it at the bottom of a lake if the bottom of the door's frame is $8.0\mathrm{m}$ deep underwater. Assume that the door is rectangular with dimensions $132\mathrm{cm}\times 87\mathrm{cm}$ and opens along the vertical axis.

A man stores small ice blocks in a well $h=4.2\mathrm{m}$ deep. To lift the ice up, he uses a heat engine between ice and the surrounding air with efficiency $\eta =12\mathrm{\%}$ of the respective Carnot engine. The temperature of available air is $T_{\mathrm{air}}=24{}^{\circ }\mathrm{C}$. How cold must the ice be at the beggining in order to retrieve it with a final temperature $T_{\max }=-9{}^{\circ }\mathrm{C}$? How is it possible even when we heat the ice up in the process?

Estimate how much the complete glaciation of an ice hockey rink costs.

Measure the dependence of the speed of swelling of sourdough on time and ambient temperature.

Consider a circuit with a coil, a capacitor, a resistor and a voltage source connected in series (i.e. they are not parallel to each other). The coil has an inductance $L$, the capacitor has a capacitance $C$ and the resistor has a resistance $R$. The voltage source creates a voltage $U=U_0\cos \left(\omega t\right)$. Assume all devices to be ideal. Using the law of conservation of energy, write the equation relating the charge, the velocity of the charge (current $I$) and the acceleration of the charge (rate of change of the current $I$). This is an equation of a damped oscillator. Compared to the equation of damped oscillations of a mass on a spring, what are the quantities analogous to mass, stiffness of the spring and friction? Find the natural frequency of these oscillations.

Furthermore, using the quantities $L$, $R$ and $\omega$, find the capacity $C$ which causes a phase shift of the voltage on the capacitor equal to $\frac{\pi }{4}$. What is the amplitude of the voltage on the capacitor, assuming this phase shift?