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mechanics of rigid bodies

(12 points)4. Series 36. Year - E. I will hang it

We have a rope wrapped around a bar with a weight of mass $m$ at one end. Measure the dependence of the mass of the weight $M$ at the other end, needed to set the rope in motion, on the number of times the rope wraps around the bar.

(10 points)3. Series 36. Year - 5. guitar

Assume you have a guitar that is perfectly tuned at room temperature. By how many semitones (in tempered tuning) will the individual strings be out of tune if we move to a campfire, where it is cooler by $10 \mathrm{\C }$? Will the guitar still sound in tune? The distance between the string attachment points is $d = 65 \mathrm{cm}$. The strings have a density $\rho = 8~900 \mathrm{kg.m^{-3}}$, a Young's modulus of elasticity $E = 210 \mathrm{GPa}$ and a thermal expansion coefficient $\alpha = 17 \cdot 10^{-6} \mathrm{K^{-1}}$.

Honza's guitar is out of tune again.

(3 points)2. Series 36. Year - 2. uncomfortable bus

figure

Jarda wanted to watch a lecture on his laptop on the bus, so he put the laptop on a flip-up shelf of the seat in front of him. The shelf has a depth of $18 \mathrm{cm}$ and is perpendicular to the vertical backrest. Jarda's laptop, which is $25 \mathrm{cm}$ wide, consists of a base weighing $1~200 \mathrm{g}$ and a screen weighing $650 \mathrm{g}$. Let us assume that both parts are of homogeneous density. What is the largest angle the laptop can open up without falling off the shelf?

Jarda is a workaholic.

(6 points)2. Series 36. Year - 3. crane on the raft

There is a raft in the middle of the river. The mass of the raft is negligible, and it carries a crane on board. The crane moves boxes of building material of mass $m$ from one river bank to another. In one cycle, the crane loads material at one side of the river, rotates to the other river bank, unloads the material there, and rotates back. Calculate the smallest value of angular displacement of the raft from its original position during one cycle. Approximate the crane by a homogenous cylinder of mass $M\_j$ and radius $r$, and a rotating jib in the shape of a slim rod of length $kr$. Assume that the velocity of the river and the „friction“ between the raft and the water are negligible.

Vojta trained as an engineer at YouTube.

(13 points)1. Series 36. Year - E. dense ice

Measure the density of ice.

Karel's previous ice-problem was rejected, so he came up with another one.

(9 points)1. Series 36. Year - P. trains

Estimate the consumption of electrical energy for one trip of the IC Opavan train. The train set consists of seven passenger cars, a 151-series locomotive and is capable of reaching a speed of $v\_{max} = 160 \mathrm{km\cdot h^{-1}}$. For simplicity, consider that all passengers are going from Prague to Opava.

The dwarf takes the train to go home.

(6 points)5. Series 35. Year - 3. under the lid

A lid has a shape of a hollow cylinder of radius $6,00 \mathrm{cm}$. The lid under which is an air of atmospheric pressure $1~013 \mathrm{hPa}$ is placed in a horizontal washbasin. While doing the dishes, we start filling the washbasin with water at room temperature. The water also gets under the lid and compresses the air trapped inside. At a certain moment, the lid starts floating. At what height is the water level at that moment? The lid weighs $200 \mathrm{g}$, its height is $2,00 \mathrm{cm}$ and negligible volume.

Danka was doing the dishes.

(7 points)5. Series 35. Year - 4. hit

The FYKOS bird plays with a baseball bat (homogeneous rod of linear density $\lambda $) and hits a baseball of mass $m$. Assume that the rod is attached at one of its ends and can rotate around that point freely. The FYKOS bird can either act on it by a constant torque $M$ or start rotating it by a constant power $P$. After completing a rotation of $\phi _0 = 180\dg $, the end of the rod hits yet motionless baseball, which results in an elastic collision. At what length of the rod $l$ does the baseball gain maximum speed? Compare both situations (i.e., constant $M$ vs. constant $P$).

Jáchym was playing with a baseball bat.

(13 points)5. Series 35. Year - E. it's already going

Measure the moment of inertia of a cylinder (regarding its main axis) and a ball (with respect to the axis passing through its center) by rolling them on an inclined plane.

Karel imagined participants rolling.

(7 points)4. Series 35. Year - 4. Analogy

Assume we have two linear springs with elastic modulus $E = 2{,}01 \mathrm{GPa}$ and a piston with viscosity $\eta = 9,8 \mathrm{GPa\cdot s}$. The dependence of stress $\sigma $ on relative extension $\epsilon $ is characterized by formula $\sigma \_s = E\epsilon \_s$ for spring, and by formula $\sigma \_d = \eta \dot {\epsilon }\_d$ for piston, where the dot represents the time derivative (Newton's notation). We connect a spring of length $l\_s$ and a piston of length $l\_d$ into series, and then we connect the other spring of length $l\_p$ in parallel to them. Abruptly, we stretch the entire system into the state of $\epsilon _0 = 0{,}2$, and we hold the extension constant. Determine, in what time (from stretching) will the stress decrease to half of the original value, if $\frac {l\_s}{l\_p} = 0{,}5$ holds.

Mirek was thinking about problems while taking an exam.

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