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## mechanics of a point mass

### 5. Series 31. Year - 3. wedge

We have two wedges with the masses $m_1$, $m_2$ and the angle $\alpha$ (see figure). Calculate the acceleration of the left wedge. Assume that there is no friction anywhere.

Bonus: Consider friction with the $f$ coefficient.

Jáchym robbed the CTU scripts.

### 5. Series 31. Year - 5. sneaky dribblet

Let's take a rounded drop of radius $r_0$ made of water of density $\rho \_v$ which coincidentally falls in the mist in the homogeneous gravity field $g$. Consider a suitable mist with special assumptions. It consists of air of density $\rho \_{vzd}$ and water droplets with an average density of $rho\_r$ and we consider that the droplets are dispersed evenly. If a drop falls through some volume of such mist, it collects all the water that is in that volume. Only air is left in this place. What is the dependence of the mass of the drop on the distance traveled in such a fog?

Bonus: Solve the motion equations.

Karal wanted to assign something with changing mass.

### 4. Series 31. Year - 3. weirdly shaped glass

We have a cylindrical glass with a small hole at the bottom of the glass. The surface area of the hole is $S$. The glass is filled with water and the water flows into a second glass by itself. The second glass has no holes. What shape should the second glass have so that the water level grows linearly inside it? The glass is supposed to have cylindrical symmetry.

Bonus: The bottom of both glasses is at the same high and the glasses are connected by the hole.

Karel was watching how the glass is being filled.

### 4. Series 31. Year - 5. impossibility of infection

Imagine that we accelerate a usually sized bacteria into velocity $v = 50 \mathrm{km\cdot h^{-1}}$ in the horizontal direction and we let it move freely in air. Estimate the distance traveled by the bacteria before it stops.

The result might be surprising for you. How is it possible to become infected this way with a bacterial infection? Discuss why is it possible despite the result.

Karel was watching TED-Ed on Youtube.

### 3. Series 31. Year - 2. small acceleration, large acceleration

In the figure, there is an ellipse with two focal points $F_1$ $F_2$ and several marked points on the ellipse. The ellipse represents a trajectory of one material point. Plot the accelerations the point experiences in given points of its trajectory. Show it in a figure. The direction and ratio of accelerations are important.

1. There is a massive body in the focal point $F_1$. The material point is orbiting it, and Kepler's $2^{\rm nd}$ law applies.
2. The absolute value of velocity of the material point is constant. It only moves along the ellipse.

### 2. Series 31. Year - 5. raining glass

A worker brought a bag of marbles to a skyscraper construction, to show off in front of his colleagues. But, what an unlucky accident – the marbles pour out and start falling through the scaffolding towards the ground. The scaffolding consists of different levels separated by height $h$. The floor of each level is made out of identical metal grid in which the holes constitute $k \%$ out of the whole grid area. Consider a simplified model of marbles falling through the scaffolding, in which if marble lands in the hole of the grid it goes through unobstructed and if it lands on the solid part of the grid its velocity drops to $0$ and starts to fall down again immediately (i.e. the size of the marbles is insignificant with respect to the size of the holes in the scaffolding and the marbles don't bounce upon landing, instead they stop and immediately roll down into a hole and continue with their fall). Ignore any potential collisions between marbles themselves. If we assume the marbles pour out of the bag with a constant mass flow of $Q$, what is the force on each level of the scaffolding, when the situation comes to a steady state?

Mirek wanted to transfer Ohm's law into mechanics.

### 0. Series 31. Year - 1.

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

### 6. Series 30. Year - 1. heavy guns

Two machine guns, that are able to shoot bullets of mass $m=25g$ and speed $v_{1}=500\;\mathrm{m}\cdot \mathrm{s}^{-1}$ with at 10 rounds per second, are attached to the front of a car. The car accelerates on a flat surface to a speed $v_{2}=80\;\mathrm{km}\cdot h^{-1}$ and then starts firing. How many shots will be fired before the car stops? The car is neutral whilst shooting, the air and tyre resistance can be ignored. The heat losses in the machine guns are also negligible.

Mirek was thinking of GTA 2.

### 5. Series 30. Year - 4. on a string

Two masses of negligible dimensions and mass $m=100g$ are connected by a massless string with rest length $l_{0}=1\;\mathrm{m}$ and spring constant $k=50\;\mathrm{kg}\cdot \mathrm{s}^{-2}$. One of the masses is held fixed and the other rotates around it with frequency $f=2\;\mathrm{Hz}$. The first mass can rotate freely around its axis. At one point the fixed mass is released. Find the minimal separation of the two masses during the resulting motion. Do not consider the effects of gravity and assume the validity of Hook's law.

### 4. Series 30. Year - 2. jerky pendulum

It is well known fact that to make a train ride as comfortable as possible, when accelerating or braking, the acceleration needs to change as little as possible. It is therefore good practice when a train starts with small, constant change of acceleration. The change of acceleration is called a jerk. Determine how does the equilibrium position of a pendulum (the angle with the vertical $φ)$. Denoting the length of the pendulum $l$, the train starts with a constant jerk $k$ ( $k=Δa⁄Δt$, where $a$ denotes acceleration) and the train is on Earth with acceleration due to gravity $g$. Bonus: Derive the equations of motion and solve them numerically for $φ(0)=0$ and $dφ⁄dt(0)=0$ for various values of $k$.

Occurred to Karel when he should have been writing his thesis.