Problem Statement of Series 6, Year 38

What is the shortest possible length a pencil can reach through sharpening? The hand can exert a maximum pressure $p$ on the pencil and the pressure is uniformly distributed over the contact area; the sharpening requires a torque $M$. The pencil is cylindrical with diameter $d$, the sharpener has length $h$, and the coefficient of static friction between the hand and the pencil is $f$. You may assume that the pencil can always be gripped optimally. Consider normal sizes of both hands and pencils only. Sharpened pencil can be defined as one with a conical tip.

Bonus: What is the maximum achievable efficiency of writing with a pencil? By efficiency, we mean the fraction of the pencil's graphite volume actually used for writing. The graphite core initially has a sharp conical tip; if not sharpened, the tip is a cylinder with height $l$ and radius $R$. The sharpener removes material so that the tip of the pencil is conical after sharpening. Assume the pencil to always be perpendicular to the paper while writing.

One way to determine the composition of a sample is mass spectrometry. Consider a time-of-flight spectrometer, in which particles (atoms and molecules) are initially ionized by removing an electron and then accelerated by a voltage of $U=10\mathrm{kV}$. They then travel at a constant velocity through the spectrometer body of length $d=2.0\mathrm{m}$ until they reach the particle detector, where their count $I$ is recorded as a function of the time of flight $t$. During ionization, a molecule may also break into several fragments, which then travel to the detector. The graph shows such a measured dependence for a known triatomic molecule made of two elements. Estimate which atoms or molecules correspond to each detected particle cluster and determine the original molecule they initially formed. Assume all charged particles entered the spectrometer body at the same time.

Bonus: With a better, more precise detector, additional particles were detected at times $9.7\mathrm{\mu }\mathrm{s}$ and $6.7\mathrm{\mu }\mathrm{s}$. How would you explain the presence of these particles if the only source was the same molecule as in the first part of the problem?

In an action move, a car is moving on a road with speed $v_0$. There is a truck traveling in front of the car at a speed $v_{\mathrm{k}}<v_0$ with its loading area opened and prepared for the car to drive into. What will be the speed of the car after its spinning wheels get onto the truck?

Assume that the front wheels of the car will be slipping for a very short time after making contact with the truck; we are interested in the speed right after the wheels stop slipping. Each car wheel has a radius $r$ and a moment of inertia $I$. The car mass is equal to $M$, while the truck mass is substantially greater. The truck will therefore see no change in speed throughout the process, and its wheels are not slipping.

A torpedo created a hole of surface area $A$ in the hull of a submarine. The hull has a volume $V_0$. Will the submarine reach the surface if hit at depth $h_0$? It can ascend toward the surface with speed $v$, the pumps are able to remove a volume of $V_1$ each second; the critical volume after which the submarine inevitably sinks to the bottom, where it is crushed by the surrounding water pressure, is equal to $V_{\mathrm{crit}}\ll V$. Assume that $v_{\mathrm{v}}$, the speed at which the water is flowing into the submarine, obeys Torricelli's law regardless of the water volume inside the submarine.

Believe it or not, there are components for which the current can decrease as the voltage increases under certain conditions. Consider a diode where, in a voltage interval between $U_1$ and $U_2$, the current decreases linearly from $I_1$ to $I_2$ (all values are positive). This diode is connected in a series with a resistor of resistance $R_0$ and a parallel RLC circuit (a resistor $R$, capacitor $C$, and inductor $L$). This circuit is then connected to a DC voltage source $U_0$. Determine the current that flows through the circuit. However, when this state is slightly perturbed, we can observe harmonic oscillations. What conditions must be satisfied, and what is the frequency of these oscillations for this to occur? Assume that the voltage and current on the diode are always within the intervals specified above.

Estimate the energy carried by Jupiter's Great Red Spot. Consider different forms of energy (kinetic, potential, chemical, etc.) and describe the various parameters that could influence this energy. Additionally, discuss how the energy levels might evolve over time.

Measure how distance traveled and velocity depend on time during a free fall. Use a light symmetric object (ideally a ping-pong ball) and drop it from a height of at least 5 meters so that the effect of air resistance becomes measurable. Compare the recorded dependencies with an appropriate theoretical model.

Hint: Record the fall on video and analyze it using suitable software, such as Tracker.

1. Using the Drude model, determine the distance traveled by an electron in copper between two collisions; the electric field intensity in the conductor equals $E=1.0\mathrm{V}\cdot {\mathrm{m}}^{-1}$. After doing so, compare the calculated value to the mean free path of electrons in copper and discuss why there is such a difference between these two values. Consider $\sigma =5.95\cdot {10}^7{\mathrm{\Omega }}^{-1}\cdot {\mathrm{m}}^{-1}$ as the electrical conductivity of copper. – 3 points

2. In figure , a graph shows a part of a cyclic voltammetry measurement of the adsorption of hydrogen on platinum. Determine the electrochemically active surface area of the platinum. A platinum monocrystal has a charge density of $240\mathrm{\mu }\mathrm{C}\cdot {\mathrm{cm}}^{-2}$. Furthermore, calculate the areal capacity and compare it to a value obtained by the Helmhlotz model introduced in the first subtask of the 4th part of the series. The scanning speed is equal to $v=15\mathrm{mV}\cdot {\mathrm{s}}^{-1}$. – 4 points

   

3. Draw a Pourbaix diagram of fykosium, a hypothetic metallic element, under standard conditions. There are three important reactions for this element: $$\mathrm{Fk}{\phantom{A}}^0(\mathrm{s})⟷\mathrm{Fk}{\phantom{A}}^{3+}(\mathrm{aq})+3\mathrm{e}{\phantom{A}}^{-}$$ with a standard reduction potential of $1.63\mathrm{V}$, $$\mathrm{Fk}(\mathrm{OH}){\phantom{A}}_3(\mathrm{s})+3\mathrm{H}{\phantom{A}}^{+}+3\mathrm{e}{\phantom{A}}^{-}⟷\mathrm{Fk}{\phantom{A}}^0(\mathrm{s})+3\mathrm{H}{\phantom{A}}_2\mathrm{O}$$ with a standard reduction potential of $2.04\mathrm{V}$, $$\mathrm{Fk}(\mathrm{OH}){\phantom{A}}_3(\mathrm{s})+3\mathrm{H}{\phantom{A}}^{+}⟷\mathrm{Fk}{\phantom{A}}^{3+}(\mathrm{aq})+3\mathrm{H}{\phantom{A}}_2\mathrm{O}(\mathrm{l}).$$ – 3 points

Bonus: Determine whether fykosium could be used as a material on an anode in a water proton exchange electrolyzer (PEM-WE). The environment has to be stable; remember that fykosium is a metallic element.