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mechanics of rigid bodies

5. Series 35. Year - 3. under the lid

A lid has a shape of a hollow cylinder of radius $6,00 \mathrm{cm}$. The lid under which is an air of atmospheric pressure $1~013 \mathrm{hPa}$ is placed in a horizontal washbasin. While doing the dishes, we start filling the washbasin with water at room temperature. The water also gets under the lid and compresses the air trapped inside. At a certain moment, the lid starts floating. At what height is the water level at that moment? The lid weighs $200 \mathrm{g}$, its height is $2,00 \mathrm{cm}$ and negligible volume.

Danka was doing the dishes.

5. Series 35. Year - 4. hit

The FYKOS bird plays with a baseball bat (homogeneous rod of linear density $\lambda $) and hits a baseball of mass $m$. Assume that the rod is attached at one of its ends and can rotate around that point freely. The FYKOS bird can either act on it by a constant torque $M$ or start rotating it by a constant power $P$. After completing a rotation of $\phi _0 = 180\dg $, the end of the rod hits yet motionless baseball, which results in an elastic collision. At what length of the rod $l$ does the baseball gain maximum speed? Compare both situations (i.e., constant $M$ vs. constant $P$).

Jáchym was playing with a baseball bat.

4. Series 35. Year - 4. Analogy

figure

Schema

Assume we have two linear springs with elastic modulus $E = 2,01 \mathrm{GPa}$ and a piston with viscosity $\eta = 9,8 \mathrm{GPa\cdot s}$. The dependence of stress $\sigma $ on relative extension $\epsilon $ is characterized by formula $\sigma \_s = E\epsilon \_s$ for spring, and by formula $\sigma \_d = \eta \dot {\epsilon }\_d$ for piston, where the dot represents the time derivative (Newton's notation). We connect a spring of length $l\_s$ and a piston of length $l\_d$ into series, and then we connect the other spring of length $l\_p$ in parallel to them (see figure ). Abruptly, we stretch the entire system into the state of $\epsilon _0 = 0,2$, and we hold the extension constant. Determine, in what time (from stretching) will the stress decrease to half of the original value, if $l\_s / l\_p = 0,5$ holds.

Mirek was thinking about problems while taking an exam.

6. Series 34. Year - 1. figure skater

Assume a figure skater, rotating around her transverse axis with her arms spread with an angular velocity $\omega $. Find her angular velocity $\omega '$, that she will rotate with her arms positioned close to her body. What work does she have to perform in order to get her arms close to her body? Finding a proper approximation of the figure skater's body is left to the reader.

Skřítek procrastinated by watching figure skating.

6. Series 34. Year - E. spilled glass

Take a glass, can or any other cylindrically symmetrical container. Measure the relationship between the angle of inclination of the container when it tips over and the amount of water inside of it. We recommend to use a container with greater ratio of its height to the diameter of its base.

Jindra was watering the table.

5. Series 34. Year - 5. rheonomous catapult

Let us have a thin rectangular panel that rotates around its horizontally oriented edge at a constant angular velocity. At the moment when the panel is in a horizontal position during rotating upwards, we place a small block on it so that its velocity with respect to the panel is zero. How will the block move on the panel if the friction between them is zero? Where do we have to place the block so that it flies away from the panel exactly after a quarter of its turn? Discuss all the necessary conditions that must be met to achieve this. Bonus: What power does the panel transfer on the block and what total work does it do on it?

Vašek was tired of problems with scleronomous bond, so he came up with rheonomous bond.

6. Series 33. Year - 3. hung

figure

What weight can be hung on the end of a coat hanger without turning it over? The hanger is made of a hook from very light wire, which is attached to the centre of the straight wooden rod, which length is $l = 30 \mathrm{cm}$ and weight $m=200 \mathrm{g}$. The hook has the shape or circular arc with radius $r=2,5 \mathrm{cm}$ and angular spread $\theta =240 \mathrm{\dg }$. The distance between the centre of the arc and the rod is $h=5 \mathrm{cm}$. Neglect every friction.

Dodo is seeking for a scarce.

5. Series 33. Year - 2. will it move?

Jachym wants to pickle cabbage at home, so he buys a cylindrical barrel. He carries it from the shop to the home using underground. We can consider the barrel and its lid as a hollow cylinder with outer dimensions: radius $r$, height $h$ and width of the walls, the base, and the lid is $t$. The barrel is made of a material with density $\rho $. What is the maximum acceleration that the underground can go with, so the free standing barrel does not move in respect to the underground? Coefficient of friction between underground's floor and the barrel is $f$.

Dodo is listening to Jachym's excuses again.

5. Series 33. Year - 3. Matěj's dream ball

Exactly on the edge of a table lies a homogenous ball with the radius $r$. Since the equilibrium is „semi-unstable“, the ball eventually starts falling off the table. What will it's angular velocity be during the fall? Assume the ball rolls without slipping.

4. Series 33. Year - E. torsional pendulum

Take a homogeneous rod, at least $40 \mathrm{cm}$ long. Attach two cords of the same material (e.g. thread or fishing line) to it, symmetrically with respect to its centre, and attach the other ends of the cords to some fixed body (e.g. stand, tripod) so that both cords would have the same length and they'd be parallel to each other. Measure the period of torsion oscillations of the rod depending on the distance $d$ of the cords, for multiple lengths of the cords, and find the relationship between these two variables. During torsion oscillations, the rod rotates in a horizontal plane and its centre remains still.

Karel wanted to hyponotize participants.

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