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mechanics of a point mass

1. Series 33. Year - S. slow start-up

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

Karel wants to have the longest problem assignment.

6. Series 32. Year - 2. bookworm

figure

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.

Vítek was in a mobile library.

6. Series 32. Year - 3. range

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.

Do not leave drills where Jáchym may take them!

6. Series 32. Year - 4. rope

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?

Matej was pulling down a climbing rope.

6. Series 32. Year - 5. elastic cord swing

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.

Matěj wonders how to hurt little children at playgrounds.

6. Series 32. Year - E. slippery

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.

Mikulas wants to go sliding.

5. Series 32. Year - 1. urban walk

Matěj walks across the street with constant velocity. Every 7 minutes a tram going in opposite direction passes, while every 10 minutes a tram going in his direction passes. We assume that trams ride in both directions with the same frequency. What is the frequency?

Matěj went for a walk

5. Series 32. Year - 5. bouncing ball

We spin a rigid ball in the air with angular velocity $\omega $ high enough parallel with the ground. After that we let the ball fall from height $h_0$ onto a horizontal surface. It bounces back from the surface to height $h_1$ and falls to a slightly different spot than the initial spot of fall. Determine the distance between those two spots of fall onto ground, given the coefficient of friction $f$ between the ball and the ground is small enough.

Matej observed Fykos birds playing with a ball

5. Series 32. Year - S. heavenly-mechanic

  1. Consider a cosmic body with the mass of five Suns surrounded by a spherically symmetrical homogenous gas cloud with the mass of two Suns and radius $1 \mathrm{ly}$. The cloud starts to collapse into the central cosmic body. Neglect the mutual interactions of particles in the cloud (excluding gravity). Find how long it will take for the whole cloud to collapse into the central body. Do not solve this problem numerically.
  2. Show that the Binet equation solves following the differential equation, which describes the motion of a mass point of mass $m$ in a spherically symmetrical central-force field. \[\begin{equation*} \dot {r}^2 = \frac {2}{m} \(E - V(r) - \frac {l^2}{2mr^2}\) \end {equation*}\] Where $r$ is the length of the radius vector, $E$ is the total energy, $l$ is angular momentum, and $V(r)$ is the potential energy of the mass point.
  3. Set up the Lagrangian for the Sun-Earth-Moon system. Assume the Sun to be motionless. The Earth and the Moon move under the influence of both the Sun and each other. While setting up the Lagrangian, think about whether you are using an appropriate number of generalized coordinates.

4. Series 32. Year - 2. it will break

Suppose a massless string of length $l$ with a point-like mass $m$ attached to its end. We know that the maximum allowed tension in the string is equal to $F = mg$, where $g$ is the gravitational acceleration. We will attach the string to the ceiling and we hold the mass in the same height with the string straight but unstrained. Then, we will release the mass and it begins to move. Find the angle (with respect to the vertical) for which the string will break.

Karel thought he won't make it.

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