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## mechanics of rigid bodies

### (8 points)2. Series 33. Year - 5. wheel with a spring

We have a perfectly rigid homogeneous disc with a radius $R$ and mass $m$, to which a rubber band is connected. It is fixed by one end in distance $2R$ from an edge of the disc and by the other end at the end of the disc. The rubber band behave as ideal, thin spring with stiffness $k$, rest length $2R$ and negligible mass. Disc is secured in the middle, so it is able to rotate in one axis around this point, but cannot move or change the rotation axis. Figure out relation between the magnitude of moment of force, by which the rubber band will be increasing or decreasing the rotation of disc depending on $\phi $. Also, figure out an equation of motion.

**Bonus:** Define the period of system's small oscillations.

### (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

### (9 points)5. Series 32. Year - P. 1 second problems

Suggest several ways to slow down the Earth so that we would not have to add the leap second to certain years. How much would it cost?

### (9 points)4. Series 32. Year - 5. frisbee

A thin homogeneous disc revolves on a flat horizontal surface around a circle with the radius $R$. The velocity of disk's centre is $v$. Find the angle $\alpha $ between the disc plane and the vertical. The friction between the disc and the surface is sufficiently large. You may work under the approximation where the radius of the disc is much less than $R$.

Jáchym hopes that contestants will come up with a solution.

### (8 points)3. Series 32. Year - 4. destruction of a copper loop

A copper flexible circular loop of radius $r$ is placed in a uniform magnetic field $B$. The vector of magnetic induction is perpendicular to the plane determined by the loop. The maximal allowed tensile strength of the material is $\sigma _p$. The flux linkage of this circular loop is changing in time as $\Phi (t) = \Phi _0 + \alpha t,$ where $\alpha $ is a positive constant. How long does it take to reach $\sigma _p$?

**Hint:** Tension force can be calculated as $T = |BIr|$.}

Vítek thinks back to AP Physics.

### (6 points)6. Series 31. Year - 3. non-analytic spring

Imagine a pole of length $b = 5 \mathrm{cm}$ and mass $m = 1 \mathrm{kg}$ and a spring of initial length $c = 10 \mathrm{cm}$, spring constant $k = 200 \mathrm{N\cdot m^{-1}}$ and negligible mass, that are connected at one of their ends. The other ends of the spring and the pole are affixed at the same height $a = 10 \mathrm{cm}$ from each other. The spring and the pole can both freely rotate about the fixed points and their joint. Label $\phi $ the angle of the pole to the horizontal. Find all angles $\phi $, for which the system is in an equilibrium. Which of these are stable and which unstable?

Jachym was supposed to come up with an easy problem.

### (7 points)6. Series 31. Year - 4. dimensional analysis

Matej was making a gun and wanted to measure what is the speed of the projectiles leaving the barrel. Unfortunately, he doesn't have any other measuring device, than a ruler. However, he found a block that is made half from steel half from wood. He lays it down at the edge of the table (of height $100 \mathrm{cm}$ and length $200 \mathrm{cm}$), and shoots at it horizontally. With the steel part of the block facing the gun, the bullet bounces off perfectly elastically and lands $50 \mathrm{cm}$ from the edge of the table. The block slides $5 \mathrm{cm}$ on the table. Then Matej turns around the block and shoots into the wooden side. This time the bullet stays in the block and the block slides only $4 \mathrm{cm}$. Help Matej with calculating the speed of the bullet. It might be also helpful to know, that when Matej lifts one edge of the table by at least $20 \mathrm{cm}$, the moving block won't stop sliding.

Matej wanted all the variables to have the same unit.

### (3 points)4. Series 31. Year - 2. autism

What is the least number of fidget spinners such that the day on Earth is extended by $1 \mathrm{ms}$ when we spin all of them? Try to guess all the missing quantities.

Matěj wants more time for spinning.

### (12 points)4. Series 31. Year - E. heft of a string

Measure the length density of the catgut which arrived to you together with the tasks. You are forbidden to weigh the catgut.

**Hint:** You can try to vibrate the string.

Mišo wondered about catguts on ITF.

### (7 points)3. Series 31. Year - 4. dropped pen

We drop a pen (rigid stick) on a table so that it makes an angle $\alpha $ with horizontal plane during its fall. Calculate the velocity of the higher end during its impact. When we dropped the pen, its center of mass was at height $h$. All collisions are inelastic and friction between the table and the end of the pen large enough.

**Bonus:** Calculate the angle $\alpha $ so that the velocity (of the second end that touches the table) is maximal. For which height $h$ is it worth to tilt the pen?

Matt was bored.