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1. Suppose we have a horizontal plane with a small hole. Through this hole goes a rope with length $l$ on which a weight of mass $M$ is hung. You may consider the weight to be a mass point. One the other end of the rope there is a second mass point with mass $m$. The rope between them is stretched thanks to the weight of mass $M$. Initially, the whole setup is in rest while the part of the rope below the plane is vertical. Then we grant the mass point on the plane velocity $v$ in a horizontal direction perpendicular to the rope as we let the system go free. Neglect all friction in this problem. Choose appropriate coordinates and find the Lagrangian for this situation.
   
2. Suppose we have an iron rod bent to a shape of a parabola given by the equation $y = x^2$. The gravity of Earth points in the negative direction of the $y$ axis. A mass point of mass $M$ can move freely along the parabola. A second mass point with mass $m$ is connected to the first by a rigid rod of length $l$. This way we have created a pendulum with a hinge sliding along the rod. The system can move only in the plane of the parabola. Find appropriate generalized coordinates and the Lagrangian for this situation.
   
3. Suppose we have line along which slides a mass point with mass $m$ (without friction). The angle between the line and the horizontal plane is $\alpha $. Find appropriate generalized coordinates and the Lagrangian for this situation. Then set up Lagrangian equations of the second kind, double-integrate them and find the solution. Do not forget about the constants of integration and explain their physical meaning. What will be their values if the mass point starts at rest at the height $h$?

How can the electronics of the Apollo landing module control an engine thrust $T$ (and so regulate the consumption of fuel), so the rocket floats onto the surface of the Moon at a steady linear motion? The effective velocity of exhaust gases is $u$. The rocket has already slowed down its motion on an orbit and goes straight down in a homogeneous gravitational field with an acceleration $g$. The initial weight of the module is $m_0$.

Bonus: How can the electronics of Apollo landing module control the engine thrust during landing from a height $h$ and initial velocity $v_0$, so the landing is so-called fall from null height and the consumption of the fuel minimalizes? Maximum engine thrust is  $T\_{max}$.

A fixed pulley is attached to the ceiling and a rope hangs over it, so the left and right end are at the same height. On one end of the rope hangs a Fykosak bird and on the other end hangs a mass, both equally heavy. Describe what happens with the system when the bird starts climbing up (on his own side of rope) with a constant force. In the beginning, assume that the rope is weightless and the pulley is ideal. Afterwards, solve this problem for a real pulley with the following parameters, its length $l$, the moment of inertia of the pulley $I$ and pulley's radius $r$. The rope's mass per unit length is $\lambda$. Assume that the rope doesn’t slip on the pulley.

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?

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.

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 $v$.

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

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.

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?

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.