Series 6, Year 32

We illuminate a mirror at an angle of $\alpha = 15\mathrm{\dg }$ with respect to the normal. We want the light to travel directly back to the source. For doing so, we can use a glass prism with an index of refraction $n = 1,8$. Find the angle $\eta $ as a function of $\alpha $ and $n$ (see the figure). The prism is placed into the air with an index of refraction $n_0$.

Hint: \[\begin{align*} \sin \(x + y\) &= \sin x \cos y + \cos x \sin y  , \\
\cos \(x + y\) &= \cos x \cos y - \sin x \sin y  , \\
\sin x + \sin y &= 2\sin \(\frac {x + y}{2}\)\cos \(\frac {x - y}{2}\)  , \\
\cos x + \cos y &= 2\cos \(\frac {x + y}{2}\)\cos \(\frac {x - y}{2}\)  . \end {align*}\]

Vítek has been spending some time in the library. Because of his clumsiness, a book fell down from a shelf and he managed to press it with a swift move towards the wall. He pushes the book with a force $F$ applied at an angle $\alpha $ (see figure). The book's mass equals $M$ and the coefficient of friction between the wall and the book is $\mu $. Find the condition under which the force keeps the book from falling down (and at rest) and determine the critical value $\alpha _0$, below which there does not exist any force that will keep the book up.

A container is filled with sulfuric acid to the height $h$. We drill a very small hole perpendicularly to the side of the container. What is the maximal distance (from the container) that the acid can reach from all possible positions of the hole? Assume the container placed horizontally on the ground.

A rope is hanging over the football goal crossbar (a horizontal cylindrical pole). When one of the rope ends is at least three times longer than the other one (the rope is hanging freely, not touching the ground), the rope spontaneously starts to slip off the crossbar. Now, we wrap the rope once around the crossbar (i.e. the rope wraps an angle of $540\dg $). How many times can the one end of the rope be longer than the other one so that the rope does not slip?

Matěj was bored by common swings, which are at playgrounds because you can swing on only forward and backwards. Therefore, he has invented his own amusement ride, which will move vertically. It will consist of an elastic cord of length $l$ attached to two points separated by distance $l$ in the same height. If he sits in the middle of the attached cord, it will stretch so that the middle will displace by a vertical distance $h$. Then, he pushes himself up and starts to swing. Find the frequency of small oscillations.

• How many cars going on the road per unit of time are needed to keep the road dry in case of raining?
• How many cars going on the road per unit of time are needed to keep the road dry (i.e. there is neither snow nor ice on the road) in case of snowing? The temperature of the snow is comparable to the surroundings (i.g. several degrees bellow zero).

Assume constant normal rate of precipitacion.

Find two plain surfaces made from the same material and measure the coefficient of friction between them. Then find out how this coefficient of friction changes when you put some free-flowing or liquid substance between them. You can use everything - water, oil, honey, melted chocolate, flour, sand, etc. Make measurements for at least 4 different substances. Discuss the results in detail and focus mainly on properties of the substances which had the greatest effect.

1. Suppose we have a simple pendulum with a mass point of mass $m$. The initial angular displacement is $120\dg $. Set up Lagrangian equations of the first kind for this pendulum and find the angular displacement for which the force acting on the pendulum rod is strongest.
2. Suppose we have a simple pendulum hanging from the mass point of another simple pendulum. The rods of both pendulums are of the same length. Set up the Lagrangian and Lagrangian equations of the second kind for this system.
3. Consider a mass point free to move along the $x$ axis, from which is hanging a simple pendulum. Find the Lagrangian of this system and using the Hamiltonian principle find the respective kinematic equations by setting the Gateaux differentials with respect to each variable equal to zero. Each Gateaux differential will then yield one kinematic equation. Compare the resultant equations to the ones you would obtain by solving this problem utilizing Lagrangian equations of the second kind.