Problem Statement of Series 2, Year 32

Your weight would be lower when the Moon is in zenith than when it is in nadir. About how much?

Imagine that Dan has a sauna with dimensions $2,5\hspace{0.17em}\mathrm{m}$ x $3\hspace{0.17em}\mathrm{m}$ x $4\hspace{0.17em}\mathrm{m}$ with a relative humidity of $20\hspace{0.17em}\mathrm{\%}$ and temperature of $90\hspace{0.17em}\text{C}$. How much water would have to evaporate, so the relative humidity inside the sauna is $35\hspace{0.17em}\mathrm{\%}$? The water evaporates inside the sauna without changing the overall temperature.

Danka won the annual Derivative Bee and she obtained a statuette made of transparent material as a reward. This statuette is made in shape of a cube prism with an edge of $a=5$ cm and height of $h\le a$. No matter what angle she looks at the prism, she can only see the reflection on the side walls but not through it. What is the index of refraction of the material? The prism is placed in air.

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  $T\mathrm{\_}max$.

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.

Create an accurate weather forecast for address V Holešovičkách 2, Prague 8, for Wednesday 14th of November from 12:00 to 15:00. How will the weather change throughout the whole day? You are allowed to use previous data about the weather in this area (remember you are only permitted to use data until 10th of November). It is necessary to justify your weather prediction, write down references and ideally to use as many data and resources as possible.

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.