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mechanics of rigid bodies

(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.

(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.

(8 points)2. Series 33. Year - 5. wheel with a spring

We have a perfectly rigid homogeneous disc with a radius $R$ and mass $m$, to which a rubber band is connected. It is fixed by one end in distance $2R$ from an edge of the disc and by the other end at the end of the disc. The rubber band behave as ideal, thin spring with stiffness $k$, rest length $2R$ and negligible mass. Disc is secured in the middle, so it is able to rotate in one axis around this point, but cannot move or change the rotation axis. Figure out relation between the magnitude of moment of force, by which the rubber band will be increasing or decreasing the rotation of disc depending on $\phi $. Also, figure out an equation of motion.

Bonus: Define the period of system's small oscillations.

Karel had a headache.

(9 points)5. Series 32. Year - 5. bouncing ball

We spin a rigid ball in the air with angular velocity $\omega $ high enough parallel with the ground. After that we let the ball fall from height $h_0$ onto a horizontal surface. It bounces back from the surface to height $h_1$ and falls to a slightly different spot than the initial spot of fall. Determine the distance between those two spots of fall onto ground, given the coefficient of friction $f$ between the ball and the ground is small enough.

Matej observed Fykos birds playing with a ball

(9 points)5. Series 32. Year - P. 1 second problems

Suggest several ways to slow down the Earth so that we would not have to add the leap second to certain years. How much would it cost?

(9 points)4. Series 32. Year - 5. frisbee

A thin homogeneous disc revolves on a flat horizontal surface around a circle with the radius $R$. The velocity of disk's centre is $v$. Find the angle $\alpha $ between the disc plane and the vertical. The friction between the disc and the surface is sufficiently large. You may work under the approximation where the radius of the disc is much less than $R$.

Jáchym hopes that contestants will come up with a solution.

(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 $T = |BIr|$.}

Vítek thinks back to AP Physics.

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