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## mechanics of a point mass

### (2 points)5. Series 25. Year - 2. electric equilibrium

An insulating rod of length $d$ and negligible mass can rotate around its middle point (see picture). There are small balls of negligible mass and charges $Q_{1}$ and 2$Q_{1}$ on both of its ends. Due to the weight $G$ (see picture) the system is in mechanical equilibrium. Distance $h$ under each of the small balls is a another ball with charge $Q$.

- What is the distance $x$ for which the rod is horizontal and is in equilibrium?
- What is the distance $h$ such that the rod is in equilibrium but there is no force on the pivot that holds it?

Dominika dug into old problem sets.

### (5 points)5. Series 25. Year - 4. mother and a stroller

Mother is connected to a stroller of mass $m$ with a string of length $l$ that is initially fully stretched. The coefficient of friction between the floor and mother resp. stroller is $f$. Mother starts pulling the stroller with constant velocity $v$ that is perpendicular to the initial orientation of the string. Describe the dependence of the trajectory of the stroller on system parameters. Assume both the mother and the stroller have negligible size. We recommend that you numerically simulate this problem.

final exam

### (4 points)5. Series 25. Year - 5. hunter v2

How much should we increase the power output of train's motor per one caught bird per second if we catch the birds in a net placed on top of the car? Train moves with speed $v$, bird's mass is $m$, his speed $w$, the angle with which the bird hits the net is $φ$ and the net has area $S$. Assume that after each catch the net returns to its initial shape.

Organizers took the train.

### (2 points)4. Series 25. Year - 1. stairs from Chrudim

Escalators at the subway station Náměstí Míru consist of $n$ steps and move with speed $v$. How many times do you actually have to step before you reach the end if you move a) in the same direction as the escalator, b) opposite to the direction the escalator moves. Assume that you move with speed $v_{1}$ such that $v_{1}>v$.

Aleš po cestě do otevřených dveří.

### (2 points)4. Series 25. Year - 2. Gatling gun

At first developed as an equipment that would simplify the distribution of seeds, gatling gun turned out to be much more useful for distributing bullets. Relative to the gatling gun's barrel, what is the region where you are in danger of being shot? The calibre is $d$, number of barrels $n$, distance from the barrel's axis to the axis of the shaft is $r$, the gun can make $f$ rotations per second, firing rate is $F$ and a bullet leaves the barrel with speed $v$.

Napadlo Michala při čtení tajného časopisu.

### (3 points)4. Series 25. Year - 3. flying stone

How long will it take for a spherical stone of mass $m$ to reach the bottom of a pond $d$ meters deep if you throw it in from height $h?$ How will the answer change if the stone is „flat“ and not spherical?

Dominika házela šutry.

### (4 points)4. Series 25. Year - 4. rockets

Model of a rocket contains a motor whose power output is constant as long as it is provided with fuel. The initial mass of the fuel is $m_{p}$, the mass of an empty rocket is $m_{0}$ and the amount of fuel burned by the motor grows linearly with time. What is the maximum height the rocket can reach assuming the gravitational field to be homogeneous and the air resistance to be negligible?

Michal odpaloval rakety.

### (2 points)3. Series 25. Year - 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.

### (4 points)3. Series 25. Year - 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.

### (8 points)3. Series 25. Year - 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).