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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

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