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## mechanics of a point mass

### (9 points)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.

### (12 points)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.

### (10 points)6. Series 32. Year - S.

- 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.
- 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.
- 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.

### (3 points)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

### (9 points)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

### (10 points)5. Series 32. Year - S. heavenly-mechanic

- 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.
- 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.
- 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.

### (3 points)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.

### (7 points)4. Series 32. Year - 4. trampoline

Two point-like masses were jumping on a trampoline into height $h_0 = 2 \mathrm{m}$. While they both were in the lowest point of the trajectory (corresponding displacement of $y = 160 \mathrm{cm}$), one of them suddenly disappeared. What is the maximum height, which was the other point-like mass bounced into? A round trampoline has circumference of $o = 10 \mathrm{m}$ and is held by $N = 42$ springs with stiffness $k = 1720 \mathrm{N\cdot m^{-1}}$. Trampoline may be modelled by $N$ springs uniformly attached around the circle and connected in the middle. Mass of the disappeared mass is $M = 400 \mathrm{kg}$.

Ivo looked after his cousin.

### (10 points)4. Series 32. Year - S.

- Show that in an arbitrary central-force field, i.e. a force field where the potential only depends on distance (not on angular position), a particle will always move in a plane.
**Instructions::**Set up Lagrangian equations of the second kind for this situation using appropriate generalized coordinates.. Then, set the coordinate $\theta = \pi /2$ and initial velocity in the direction of this coordinate equal to zero. Think about and explain why this choice of coordinates does not cause a loss of generality. - Set up the Lagrangian for a mass point moving in a plane in a central-force field. Find all the integrals of motion for this Lagrangian and use them to find the first orded differential equation for the variable $r$.
- Think about how to find the angular distance between two points on a sphere, given their spherical coordinates. Check your solution on the stars Betelgeuse and Sirius.
**Hint::**This problem can be easily solved even without the knowledge of spherical trigonometry.

### (8 points)3. Series 32. Year - 5. pointy

Consider a point-like particle in one-dimensional space. Initially, the particle is in rest at the origin of coordinates. It can be moved with acceleration from interval $\left (- a , a\right )$. Let $M\left (t\right )$ be a set of all possible physical states $\left (x, v\right )$ of positions $x$ and velocities $v$, which particle can achieve after time $t$ is elapsed. If we plot set $M\left (t\right )$ into $v(x)$ coordinate system we get surface $S\left (t\right )$. Find analytic expression for boundaries of $S\left (t\right )$.

**Bonus:** Find area of $S\left (t\right )$ as a funcion of time.}

Jáchym wanted to solve a certain trivial problem as a special case of this one.