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## mechanics of rigid bodies

### (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 $T = |BIr|$.}

### (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.

### (6 points)1. Series 31. Year - 3. hung L

We have a thin homogeneous pipe in the shape of an L with segments of lengths $b,c$. It is hung freely by one of the ends in a rail car so that its bend points in the direction of travel. Find the acceleration of the rail car required for the bottom side of the L to stay parallel to the direction of travel? Ignore relativistic effects.

*Bonus:* Consider relativistic effects.

Author is unknown, they probably hanged themselves.

### (12 points)1. Series 31. Year - E. skewer elasticity

Measure the bending of a wooden skewer stick freely supported at its ends as a function of the force applied in its middle (see picture).

Mišo was watching a crane.

### (8 points)0. Series 31. Year - 5.

### (7 points)6. Series 30. Year - 4. shoot your rat

Mirek wants to shoot a rat he sees at the dorm. To that end, he made a simple air gun which can be modeled as a tube with constant cross-section $S=15\;\mathrm{mm}$ and length $l=30\;\mathrm{cm}$ closed on one side and open on the other. Mirek plans to place a bullet of mass $m=2g$ into the tube so that the bullet seals to tube exactly and is fixed at a distance $d=3\;\mathrm{cm}$ from the closed end. He that pumps up the closed section to a pressure $p_{0}$ and then releases the bullet. He wants the speed of the bullet to be at least $v=90\;\mathrm{m}\cdot \mathrm{s}^{-1}$ as it exits the tube. What pressure will he need to achieve if the gas is ideal? Discuss the realism of the situation. Assume the bullet is released by a quasi-static adiabatic process where $κ=7⁄5$, as the gas is diatomic. Assume an external atmospheric pressure $p_{a}=10^5Pa$. Neglect losses due to friction, air resistance and gas compression ahead of the bullet.

Karel wanted to find out if the solvers could pass the Masters programme admissions at MFF