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## mechanics of rigid bodies

### (3 points)3. Series 37. Year - 2. stable sheep

Consider a rectangular board and a block of wood with dimensions $a=20 \mathrm{cm}$, $b=10 \mathrm{cm}$, and $c=5 \mathrm{cm}$ (the shape of the inverted letter $L$ is our approximation of a sheep). The edges of the board are parallel to the edges of the base of the block. Assuming the block tips over before sliding, at what angle will it tip over if we tilt it successively around each of the edges of the board? (See figure)

### (8 points)3. Series 37. Year - 4. size matters

A sphere with a radius $r$ rolls on a horizontal surface with a speed $v$. However, its path is blocked by a perpendicular step with the height $h$. Find the conditions under which the ball rolls onto the step and starts rolling along it without losing contact with the step. Under these conditions, determine its speed after it has crossed the step. Assume that all collisions are perfectly inelastic and the friction between the ball and the step is high. The step is angular and is oriented perpendicular to the direction of the sphere's motion.

### (3 points)2. Series 37. Year - 2. inflated tyre

It is said that if you want to inflate a car's tires, you should do it when they are cold. Therefore, Jarda drove to a petrol station with a compressor, bought a hot dog, and waited for the tires to cool down. Curious, he measured the tire pressure before and after his snack. It had dropped from $2{,}7 \mathrm{bar}$ to $2{,}5 \mathrm{bar}$. He wondered whether the tire pressure could be determined by the height of the car's body above the road. How much did the body of Jarda's car approach the ground due to the decrease in the tire temperature? The weight of the car is $1{,}3 \mathrm{t}$. The outer radius of the tires is $32 \mathrm{cm}$, the inner radius is $22 \mathrm{cm}$, and their width is $21 \mathrm{cm}$. Assume that the tires deform due to the car's weight only on the underside where they touch the ground.

Jarda would drain his soul (as well as his bicycle's inner tube) for FYKOS.

### (10 points)2. Series 37. Year - 5. ferry

Imagine a ferry in the shape of a rectangular cuboid with a weight $M$, length $L$, width $W$, and height $H \ll L$ from the keel to the deck. After docking at the pier, passengers gradually exit through the back of the deck so that the empty front part of the deck becomes larger and the area density of people on the filled part does not change in a different way. Find the maximum weight of passengers the ferry can carry so that no part of the deck is below the surface when people disembark. Consider that the ship is stable in the transverse direction and that people get off slowly.

After quite some time, Dodo was at sea again.

### (5 points)1. Series 37. Year - 3. new bicycle

A cyclist with the mass $m_c=62{,}3 \mathrm{kg}$ started riding his bike at constant power from rest to the wanted speed at time $t=103 \mathrm{s}$. His bicycle's steel frame and fork have a mass $M=6{,}50 \mathrm{kg}$, and each of the two wheels has a mass $m=1950 \mathrm{g}$. How long would it take him to get going on a bike with a carbon frame and fork that is four times lighter? The weight of the other bicycle parts is included in the cyclist's weight.

Dodo borrowed his sister's bike.

### (7 points)1. Series 37. Year - 4. truck flip

Legolas had a dream in which the truck braked so quickly that the container lifted off the ground and did a somersault over the cab. He wondered if that was possible, so he tried to do the math. In his model, the entire truck has a mass of $m$ and comprises a tractor and a container. It can rotate freely in all directions around the point where it is connected to the tractor. When the truck is on a flat road, the center of gravity of the container is $h$ above this connection and at a distance $l$ from it. Depending on the slope of the road $\phi$, how much force must the truck brake in order to lift the wheels under the container off the road?

Lego was literally dreaming

### (10 points)5. Series 36. Year - P. tense tree

Describe as many natural influences as possible that cause uprooting/severe damage to a lone tree in a meadow. Try to analyze one of them qualitatively as best as you can. What is the difference between a broadleaved tree and a conifer?

Bonus: Discuss some of the influences quantitatively.

Danka watched the trees bend in the wind.

### (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.

Patrik thinks about different methods of… calculation.

### (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

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? 