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## mechanics of rigid bodies

### (3 points)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.

### (12 points)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.

### (10 points)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.

### (3 points)4. Series 34. Year - 2. there is always another spring

Find the work needed to twist a spring from equilibrium position to an angular displacement of $\alpha =60\dg$. We are holding the spring in the twisted position with a torque $M=1{,} \mathrm{N\cdot m}$.

Dodo was hanging laundry on a string.

### (5 points)6. Series 33. Year - 3. hung

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.

### (3 points)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.

### (6 points)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.

### (12 points)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.

### (12 points)3. Series 33. Year - E. dense measurement

Construct a hydrometer (for example from straw and plasteline) and measure dependence of water density on the concetration of salt dissolved in it.

Bouyant Matěj.

### (6 points)2. Series 33. Year - 3. Danka's (non-)equilibrium cutting board

Cutting board with thickness $t=1,0 \mathrm{mm}$ and width $d =2,0 \mathrm{cm}$ is made up of two parts. The first part has density $\rho _1 =0,20 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_1 = 10 \mathrm{cm}$, the second part has density $\rho _2 =2,2 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and length $l_2 = 5,0 \mathrm{cm}$. We place the cutting board on water surface, which density is $\rho \_v = 1{,}00 \cdot 10^{3} \mathrm{kg\cdot m^{-3}}$ and then we wait until it is in equilibrium position. What angle will a plane of the cutting board hold with the water surface? How big the part of the cutting board which will stay above the water level will be?

Danka was talking with Peter about dish-washing.

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