Problem Statement of Series 2, Year 40

Jirka wanted to perform a simple experiment with a solar panel with a terminal voltage of $U=12\,\mathrm{V}$ and a maximum power of $P_0=6\,\mathrm{W}$. However, he found that his resistors can handle only $P_{\mathrm{max}}=1/4\,\mathrm{W}$. Design a circuit such that no component is damaged while the solar panel is used to its maximum capacity.

During the exam period, Honza decided to calm himself with chocolate. He bought four identical homogeneous chocolate bars of length $x$ and height $h$. He stacked them as follows: the top and bottom bars are aligned, and each of the two middle bars is shifted by $x/3$ to the right relative to the bar above it, as illustrated in the attached figure.

Will the chocolate tower assembled in this way be in equilibrium? If so, will this equilibrium be stable or unstable? The bars are only freely placed on top of one another. Each bar has the same mass.

Jirka's balloon flew away, which gave him the idea that aluminum could be transported by constructing a sealed vessel filled with hydrogen. What dimensions should this vessel have if we have $1\,\mathrm{t}$ of aluminum available and, for strength, the wall should be as thick as possible? Assume hydrogen at atmospheric pressure and room temperature. Use the densities of aluminum $\rho_{\mathrm{Al}}=2700\,\mathrm{kg\!\cdot\! m^{-3}}$, hydrogen $\rho_{\mathrm{H}}=0.09\,\mathrm{kg\!\cdot\! m^{-3}}$, and air $\rho_{\mathrm{air}}=1.2\,\mathrm{kg\!\cdot\! m^{-3}}$.

The problem of the oscillation period of a body released into a tunnel through the center of the Earth is well known. But what if we dig our tunnel not through the center of the Earth, but at a specified distance $d$ from the center? What will the oscillation period be, that is, the time it takes the released body to travel from one end of the tunnel to the other and back? Assume a zero coefficient of friction between the tunnel wall and the body and a tunnel of very small width. Assume that the presence of the tunnel does not affect the Earth's gravitational field.

Bonus: How does the motion of the point mass change if we take its friction with the wall into account? What distance does it travel before coming to rest?

Jarda is lying on a bed with a nightstand beside him, on which his watch is lying face up. Jarda's eyes are $H=20\,\mathrm{cm}$ below the level of the nightstand and at a horizontal distance $d=15\,\mathrm{cm}$ from its edge. The watch lies $L=30\,\mathrm{cm}$ from the edge of the nightstand.

Between the watch and the edge of the nightstand stands a transparent glass in the shape of a rectangular cuboid with a square base of side length $a=6\,\mathrm{cm}$. The center of the bottom of the glass is located exactly $x=16\,\mathrm{cm}$ from the edge of the nightstand. To what height must the glass be filled with water for Jarda to see exactly the center of the watch? Consider the light ray that just grazes the edge of the nightstand. Neglect the thickness of the glass walls.

This problem has an open solution, so be sure to cite all sources used.

Estimate the distance at which noise from a highway can still be heard and beyond which it is possible to live comfortably. How does the surrounding environment affect this distance? How does the situation change if a noise barrier is built along the highway?

Measure the calorific value of plain paper.