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## mechanics of a point mass

### (5 points)6. Series 33. Year - 3. hung

What weight can be hung on the end of a coat hanger without turning it over? The hanger is made of a hook from very light wire, which is attached to the centre of the straight wooden rod, which length is $l = 30 \mathrm{cm}$ and weight $m=200 \mathrm{g}$. The hook has the shape or circular arc with radius $r=2,5 \mathrm{cm}$ and angular spread $\theta =240 \mathrm{\dg }$. The distance between the centre of the arc and the rod is $h=5 \mathrm{cm}$. Neglect every friction.

### (3 points)5. Series 33. Year - 1. train on a bridge

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 $m = 574 \mathrm{t}$.

Danka watched trains from her dormitory.

### (3 points)5. Series 33. Year - 2. will it move?

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 $f$.

Dodo is listening to Jachym's excuses again.

### (10 points)5. Series 33. Year - S. min and max

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

### (3 points)4. Series 33. Year - 1. tchibonaut

Consider an astronaut of weight $M$ remaining still (with respect to a space station) in zero-g state, holding a heavy tool of weight $m$. The distance between the astronaut and the wall of the space station is $l$. Suddenly, he decides to throw the tool against the wall. Find his distance from the wall when the tool hits it.

Karel wanted to set this name for this problem.

### (3 points)4. Series 33. Year - 2. Mach number

Planes at high flight levels are controlled using the Mach number. This unit describes velocity as a multiple of the speed of sound in the given environment. However, the speed of sound changes with height. What is the difference in the speed of a plane, flying at Mach number $0{,}85$, at two different flight levels FL 250 ($7\;600 \mathrm{m}$) and FL 430 ($13\;100 \mathrm{m}$)? At which flight level is the speed higher and by how much (in $\jd {kph}$)? The speed of sound is given by $c =$331{,}57+0{,}607\left \lbrace t \right \rbrace$ \jd {m.s^{-1}}$, where $t$ is temperature in degrees Celsius. Assume a standard atmosphere, where temperature decreases with height from $15 \mathrm{\C }$ by $0,65 \mathrm{\C }$ per $100 \mathrm{m}$ (for heights between $0$ and $11 \mathrm{km}$) till $-56{,}5 \mathrm{\C }$, and then remains constant till $20 \mathrm{km}$ above mean sea level.

Karel was learning Air Traffic Control.

### (9 points)4. Series 33. Year - 5. a shortcut across time

Jachym is located in a two dimensional Cartesian system at a point $J = (-2a, 0)$. As fast as possible, he wants to get to a point $T = (2a, 0)$, which is located (luckily) in the same system. Jachym moves exclusively with velocity $v$. This is not so easy, because there is a moving strip in the shape of a line passing through points $(-3a, 0)$ and $(0, a)$. On the moving strip, Jachym is moving with total velocity $kv$. For what minimum $k \ge 1$ is it profitable for Jachym to get on the moving strip?

Jachym, from life experience.

### (8 points)3. Series 33. Year - 4. ladybird on a rubber

Ladybird moves with velocity $4 \mathrm{cm\cdot s^{-1}}$. When we place the ladybird onto a rubber, she comes through it in $10 \mathrm{s}$. What happens when the ladybird starts moving and we start prolonging the rubber the way that its length will be increasing with velocity $5 \mathrm{cm\cdot s^{-1}}$? Is the ladybird able to come through the whole rubber to its end? If yes, how long will it take? Consider that the rubber prolongs uniformly and never breaks.

Matej was watching Vsauce.

### (10 points)3. Series 33. Year - S.

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

### (10 points)2. Series 33. Year - S.

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