Problem Statement of Series 2, Year 33

They say that people inside an elevator can bear with acceleration $a=2.50\mathrm{m}\cdot {\mathrm{s}}^{-2}$ without any major problems. We want to get to the planned floor as soon as possible. If the elevator started running with that acceleration for a quarter of a time, a half of the time went by constant velocity, and the rest quarter of the time it was decelerating, how high it could go by total time of the ride $t=1.00\min$?

Let us assume a pulley in a fixed position with a rope of negligible weight. A weight $m_1$ is located on one end of the rope and a winch (mass $m_2$) on the other. Initially, the winch is in the same height as the weight $m_1$. In the first case, the winch is fixed to the ground and pulls only the weight. In the second case, the weight is firmly connected to the winch. Therefore, while the rope is being attached, both the weight and the winch are pulled up. Find out which case requires less pull force (and therefore weaker winch).

Cutting board with thickness $t=1.0\mathrm{mm}$ and width $d=2.0\mathrm{cm}$ is made up of two parts. The first part has density ${\rho }_1=0.20\cdot {10}^3\mathrm{kg}\cdot {\mathrm{m}}^{-3}$ and length $l_1=10\mathrm{cm}$, the second part has density ${\rho }_2=2.2\cdot {10}^3\mathrm{kg}\cdot {\mathrm{m}}^{-3}$ and length $l_2=5.0\mathrm{cm}$. We place the cutting board on water surface, which density is ${\rho }_{\mathrm{v}}=1.00\cdot {10}^3\mathrm{kg}\cdot {\mathrm{m}}^{-3}$ and then we wait until it is in equilibrium position. What angle will a plane of the cutting board hold with the water surface? How big the part of the cutting board which will stay above the water level will be?

We have a perfectly rigid homogeneous disc with a radius $R$ and mass $m$, to which a rubber band is connected. It is fixed by one end in distance $2R$ from an edge of the disc and by the other end at the end of the disc. The rubber band behave as ideal, thin spring with stiffness $k$, rest length $2R$ and negligible mass. Disc is secured in the middle, so it is able to rotate in one axis around this point, but cannot move or change the rotation axis. Figure out relation between the magnitude of moment of force, by which the rubber band will be increasing or decreasing the rotation of disc depending on $\varphi$. Also, figure out an equation of motion.

Bonus Define the period of system's small oscillations.

Estimate about how much would a content of $\mathrm{CO}{\phantom{A}}_2$ in the atmosphere rise, if all flora on Earth was burnt down?

Measure your volume using several different methods.