Assignment of Series 3 of Year 37

Danka has a humidifier in her dorm room, which evaporates water from its boiling point to create warm steam. The device can hold a maximum of $V = 3.8\,\mathrm{l}$ of water, which it uses up in $t = 24\,\mathrm{h}$. What is its efficiency, i.e., what fraction of the energy drawn from the electrical grid it uses to convert the water to steam? The input power of the humidifier is $P = 260\,\mathrm{W}$, and Danka put water at $T_0 = 20\,\mathrm{^\circ\mskip-2mu\mathup{C}}$ inside. All the necessary properties of water can be looked up.

In the microworld of cells, there are two types of transport: transport by free diffusion, also known as Brownian motion where the motion uses the energy of the environment directly. The second type, so-called active transport, requires, among other things, a motor protein moving at a constant speed along the cytoskeletal filament. Consider the typical value of the diffusion constant $D \approx 10^{-9}\,\mathrm{cm^2\cdot s^{-1}}$ and the rate of active transport speed $u\approx10^{-6}\,\mathrm{m\cdot s^{-1}}$. For which distances is the Brownian motion more time efficient than the active transport? Assume that the transport is happening in only one direction.

A sphere with a radius $r$ rolls on a horizontal surface with a speed $v$. However, its path is blocked by a perpendicular step with the height $h$. Find the conditions under which the ball rolls onto the step and starts rolling along it without losing contact with the step. Under these conditions, determine its speed after it has crossed the step. Assume that all collisions are perfectly inelastic and the friction between the ball and the step is high. The step is angular and is oriented perpendicular to the direction of the sphere's motion.

Assume a cylindrical glass of negligible mass, internal cross-sectional area $S$, and height $h$ that is turned upside down and its open rim aligned with the water level in the reservoir. We start to push slowly downwards. What work will we perform if we move the jar with the air inside so that its base $d>0$ is below the surface?

Bonus: Let us now consider a more realistic case. How much work must be performed to completely submerge a jar of the same dimensions but mass $m$ to the bottom of a container with area $A$ and initial water level in height $H$? Assume that the jar is completely submerged when it reaches the bottom.

What determines the width of a lightning channel in a thunderstorm? Create a quantitative model.

Attach a string at two points at a fixed distance $L$ and ensure it is always taut during measurement. Determine the dependence of the fundamental frequency of its oscillations on temperature.

1. According to definitions by International System of Units, convert these into base units
   • pressure $1\,\mathrm{psi}$,
   • energy $1\,\mathrm{foot-pound}$,
   • force $1\,\mathrm{dyn}$.
2. In the diffraction experiment, table salt's grating constant (edge length of the elementary cell) was measured as $563\,\mathrm{pm}$. We also know its density as $2.16\,\mathrm{g\cdot cm^{-3}}$, and that it crystallizes in a face-centered cubic lattice. Determine the value of the atomic mass unit.
3. A thin rod with a length $l$ and a linear density $\lambda$ lies on a cylinder with a radius $R$ perpendicular to its axis of symmetry. A weight with mass $m$ is placed at each end of the rod so that the rod is horizontal. We carefully increase the mass of one of the weights to $M$. What will be the angle between the rod and the horizontal direction? Assume that the rod does not slide off the cylinder.
4. How would you measure the mass of:
   • an astronaut on ISS,
   • a loaded oil tanker,
   • a small asteroid heading towards Earth?