# 3. Series 25. Year

### (2 points)1. Watt's regulator

On attached picture you can see a system consisting of two heavy balls attached to a joint using two rigid rods. Assume that the motion of the balls is constrained to a certain vertical plane. The whole system starts to rotate around a vertical axis passing through the joint. How does the angle $α$ depend on the angular velocity of the system $ω?$

Aleš po cestě do otevřených dveří.

### (2 points)2. river ride

Lock is placed in a dam on a river in order for ships to be able to pass through. Assume the river has a flow rate of $Q=200m⁄s$ and that the lock connects places with height difference $H=4\;\mathrm{m}$ and has dimensions $s=100\;\mathrm{m}$, $d=20\;\mathrm{m}$. How many ships per day can this lock transport from the lower place to the higher place if the maximum flow rate in and out of the lock is $Q_{Z}=250m⁄s?$

Napadlo Michala při čtení tajného časopisu.

### (4 points)3. train à grande vitesse

Railway has a shape of an arc of radius $R$ and its width is $D$. Train's center of mass is at height of $H$ meters. How fast can a train go on this railway if, no matter where it stops, it never falls? Under what conditions is this maximum speed unbounded? Note Neglect forces cars act on each other and also assume the width of a car is much smaller than the radius $R$.

Dominika házela šutry.

### (4 points)4. cutting down a tree

There can be many problems when cutting down a tree. Imagine a rod attached to an unstretchable rope placed on a pulley. Two workers are standing under the tree to make sure the rod does not fall into a pool. The rod falls down a distance $h$ before the rope is tightened. Under certain conditions the workers that hold the rope can be puled so high that they hit the pulley. Give conditions under which cutting of this branch is safe.

Hint: First consider two masses on ice connected by an unstretchable rope and having distinct velocities.

Michal odpaloval rakety.

### (5 points)5. gas leakage

Imagine two infinitely large grounded conducting planes distance $l$ apart. There is a charge placed between them and distance $x$ from one of them. What is the induced charge on the other plane?

### (5 points)P. save the world

Invent a mechanism that converts rotational energy of the Earth to electric energy. Do not be too down-to-earth. Everything is possible.

Pikoš platil účet za plyn.

### (8 points)E. bouncing of a ball

Roll a very flexible ball perpendicularly against a wall. When it bounces off the wall it jumps. How does the distance of the point where the ball hits the ground depend on the initial velocity of the ball? You can also investigate dependence on other parameters.

Note You can find some useful information in the text on our web.% (http://fykos.cz/rocnik25/3-e_std-text.pdf).