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## mechanics of rigid bodies

### (13 points)1. Series 36. Year - E. dense ice

Measure the density of ice.

### (9 points)1. Series 36. Year - P. trains

Estimate the consumption of electrical energy for one trip of the IC Opavan train. The train set consists of seven passenger cars, a 151-series locomotive and is capable of reaching a speed of $v\_{max} = 160 \mathrm{km\cdot h^{-1}}$. For simplicity, consider that all passengers are going from Prague to Opava.

### (6 points)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.

### (7 points)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.

### (13 points)5. Series 35. Year - E. it's already going

Measure the moment of inertia of a cylinder (regarding its main axis) and a ball (with respect to the axis passing through its center) by rolling them on an inclined plane.

Karel imagined participants rolling.

### (7 points)4. Series 35. Year - 4. Analogy

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. 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 $\frac {l\_s}{l\_p} = 0{,}5$ holds.

Mirek was thinking about problems while taking an exam.

### (12 points)4. Series 35. Year - E. Useful Coin

Measure at least three physical properties of the smallest coin of legal tender in your country. We consider macroscopic dimensions as one property. We evaluate not only the accuracy of the measurement and the detail of the description but also the originality in the selection of quantities.

Karel wanted the participants to observe the money.

### (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.