Problem Statement of Series 5, Year 33

There is a freight train standing on a $300\mathrm{m}$ long bridge. The mass of the train is evenly distributed onto area of all nine steel pillars of the bridge. Every pillar has a base in a shape of a square with a side $a=2.0\mathrm{m}$ and a height $h=10\mathrm{m}$. How much do the steel pillars shrink under the weight of the train? Young modulus of steel is $E=200\mathrm{GPa}$. Overall mass of the train is $m=574\mathrm{t}$.

Jachym wants to pickle cabbage at home, so he buys a cylindrical barrel. He carries it from the shop to the home using underground. We can consider the barrel and its lid as a hollow cylinder with outer dimensions: radius $r$, height $h$ and width of the walls, the base, and the lid is $t$. The barrel is made of a material with density $\rho$. What is the maximum acceleration that the underground can go with, so the free standing barrel does not move in respect to the underground? Coefficient of friction between underground's floor and the barrel is $f$.

Exactly on the edge of a table lies a homogenous ball with the radius $r$. Since the equilibrium is “semi-unstable”, the ball eventually starts falling off the table. What will it's angular velocity be during the fall? Assume the ball rolls without slipping.

A circular metal loop of mass $m=18\mathrm{g}$, radius $r=15\mathrm{cm}$ and electrical resistance $R=3.5\mathrm{m}\mathrm{\Omega }$ is at rest. By the resistance of a loop we mean resistance between the ends of a wire created by cutting the loop in one place. At the time $t=0$ we create a homogenous magnetic field perpendicular to the plane of the loop. The magnetic field strength changes as a function of time $B(t)=\alpha t$, where $\alpha =1\mathrm{mT}\cdot {\mathrm{s}}^{-1}$ is a constant. Because of the nonstationary magnetic field, the loop will start to turn slowly around it's axis. Calculate the angular velocity $\omega$ at time $t=0.1\mathrm{s}$. Neglect the deformation of the loop.

Calculate the phase shift $\mathrm{\Delta }\mathrm{\Phi }$ when an optical beam with a wavelength ${\lambda }_0$ goes through a glass plate with thickness $h$ and the index of refraction $n$ that is moving along the beam with constant speed $v$ relative to a case when the plate is stationary. We are interested mainly about the first nonzero term of Taylor series of $\mathrm{\Delta }\mathrm{\Phi }(v)$.

Estimate the time that passes between the flip of a light switch and the turning on of the light source. Make independent estimates for a light bulb, fluorescent lamp, LED light bulb and Neon tube light. Discuss as many factors influencing the time as you can.

Measure the time dependence of the temperature of a liquid in an open mug. Use water first, than oil and finally water with a thin layer of oil. The layer should be as thin as possible but still should cover the whole surface. Measure between $90{}^{\circ }\mathrm{C}$ and $50{}^{\circ }\mathrm{C}$. Be careful to keep all conditions same for all experiments (the same mug, the same initial temperature, keep the thermometer on the same place in the liquid etc.). Describe your experimental equipment, compare cooling in individual cases and discuss the result.