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mechanics of a point mass

(12 points)2. Series 32. Year - E.

Measure an average vertical velocity of falling leaves. Use leaves from several different trees and discuss what impact the shape of a leaf has on the velocity. How should an ideal leaf look like when we want it to fall as slow as it is possible?

Jachym got this idea, when he asked his friend, whether he knew any interesting experiment

(10 points)2. Series 32. Year - S.

  1. Suppose we have a dumbbell consisting of two mass points with masses $m$ and $M$ connected via a massless rod. This dumbbell is in a free fall. Write a constraint function and Lagrangian equations of the first kind for this object.
  2. Suppose we have a triangular prism with mass $M$ on a horizontal platform as in the picture. A mass point with the mass $m$ is sliding down a side of the prism. The angle between said side and the platform is $\alpha $. You may neglect friction.
  • Set up Lagrangian equations of the first kind for this situation.
  • Show that, for zero initial speed of the mass point, the total momentum of this system in the direction of $x$ axis is zero.
  • Solve the system of (Lagrangian) equations and find the time-dependent equations for the speeds of the prism and the mass point.
  • Find the ratio between these two speeds.


  1. Set up Lagrangian equations of the first kind for a simple pendulum. Show that the law of conservation of energy holds for this situation.

(3 points)1. Series 32. Year - 2. fireworks

Jachym was launching fireworks. We can imagine it as a beacon, that is at some point shot straight up with velocity $v$, and explodes after a certain delay. Jachym was standing a distance $x$ from the launch site when he heard the launch of the fireworks. After a delay $t_1$ he saw the explosion and after another delay $t_2$ he heard the explosion. Calculate the velocity $v$.

Jachym can't hide his pyrotechnic affinity.

(7 points)1. Series 32. Year - 4. Skyfall

When James Bond let go of agent 006 Alec Treveljan from the top of the Arecibo radiotelescope in the final scene of the film Golden Eye, the falling agent started screaming with a frequency $f$. How does the frequency agent 007 hears at the top of the telescope change as a function of time. Neglect air resistance.

Hint: Ask Mr. Doppler

Matej enjoys looking outside

(10 points)1. Series 32. Year - S. theoretical mechanics

Before we dive into the art of analytical mechanics, we should brush up on classical mechanics on the following series of problems.

  1. A homogenous marble with a very small radius sits on top of a crystal sphere. After being granted an arbitrarily small speed, the marble starts rolling down the sphere without slipping. Where will the marble separate and fall of the sphere?
  2. Instead of the sphere from the previous problem, the marble now sits on a crystal paraboloid given by the equation $y = c - ax^2$. Again, where will the marble separate from the paraboloid?
  3. A cyclist going at the speed $v$ takes a sharp turn to a road perpendicular to his original direction. During the turn, he traces out a part of a circle with radius $r$. How much does the cyclist have to lean into the turn? You may neglect the moment of inertia of the wheels and approximate the cyclist as a mass point.
    Bonus: Do not neglect the moment of intertia of the wheels.

(8 points)6. Series 31. Year - 5. jump from a plane

Filip of mass $80 \mathrm{kg}$ jumped out of air plane, that is $h_1 =500 \mathrm{m}$ above the ground. At the same time, Danka (mass $50 \mathrm{kg}$) jumps out of a different airplane but from a height of $h_2 =569 \mathrm{m}$. Assume both of them have the same drag coefficient $C = 1{,}2$, Filip's cross-sectional area is $S\_F = 2{,}2 \mathrm{m^2}$ and Danka's is $S\_D=1{,}5 \mathrm{m^2}$. The density of air $\rho =1{,}205 \mathrm{kg\cdot m^{-3}}$ is and stays the same in all heights. At what time will Danka be at the same height above the ground as Filip?

Danka contemplated the strenuous life of a physicist and wanted to break free for a moment.

(3 points)5. Series 31. Year - 1. staircase on the Moon

If we once colonized the Moon, would it be appropriate to use stairs on it? Imagine the descending staircase on the Moon. The height of one stair is $h=15 \mathrm{cm}$ and it's length is $d=25 \mathrm{cm}$. Estimate the number $N$ of stairs that a person would fly over if he walked into the staircase with a velocity $v=5{,}4 \mathrm{km\cdot h^{-1}}=1{,}5 \mathrm{m\cdot s^{-1}}$. The gravitational acceleration on the Moon's surface is six times weaker than on Earth's surface.

Dodo read The Moon Is a Harsh Mistress.

(5 points)5. Series 31. Year - 3. wedge

figure

Situation

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.

(8 points)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.

(6 points)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.

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