# Series 2, Year 33

* Upload deadline: 19th November 2019 11:59:59 PM, CET (local time in Czech Republic)*

### (3 points)1. fast elevator

They say that people inside an elevator can bear with acceleration $a = 2{,}50 \mathrm{m\cdot s^{-2}}$ 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 $t = 1{,}00 \mathrm{min}$?

Karel rides an elevator.

### (3 points)2. weak winch

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).

Vašek needed a mechanism to pull up a snow plow blade.

### (6 points)3. Danka's (non-)equilibrium cutting board

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 $\rho _1 =0,20 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_1 = 10 \mathrm{cm}$, the second part has density $\rho _2 =2,2 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_2 = 5,0 \mathrm{cm}$. We place the cutting board on water surface, which density is $\rho \_v = 1{,}00 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ 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?

Danka was talking with Peter about dish-washing.

### (7 points)4. butterflies

A rainbow-like, turquoise colour of the genus Morpho butterflies' wings' surface is a consequence of constructive interference of light reflected from layers of the transparent cuticle (cell layer on the wings' surface). Layers of thickness $h\_t = 63{,}5 \mathrm{nm}$ and refractive index $n\_t = 1{,}53$ are separated by $h\_a = 120{,}3 \mathrm{nm}$ thick air gaps (see figure). Estimate the wavelengths of visible light corresponding to interference maxima.

Domča was catching butterflies during the exam period in January.

### (8 points)5. wheel with a\protect \unhbox \voidb@x \penalty \@M \ {}spring

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 $\phi $. Also, figure out an equation of motion.

**Bonus:** Define the period of system's small oscillations.

Karel had a headache.

### (10 points)P. Earth fired up

Estimate about how much would a content of $\ce {CO2}$ in the atmosphere rise, if all flora on Earth was burnt down?

Karel is pyromaniac.

### (13 points)E. I need a\protect \unhbox \voidb@x \penalty \@M \ {}hug

Measure your volume using several different methods.

Matej was having a bath.

### (10 points)S.

*We are sorry. This type of task is not translated to English.*