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mechanics of rigid bodies

(5 points)3. Series 25. Year - P. save the world

Invent a mechanism that converts rotational energy of the Earth to electric energy. Do not be too down-to-earth. Everything is possible.

Pikoš platil účet za plyn.

(4 points)2. Series 25. Year - 3. lifting boats

figure

A small Scottish town decided to build an elevator for boats. It consists of two big containers filled with water that are suspended from the ends of a long rod. This rod is attached in the middle to a motor that can rotate the whole system. A boat enters one of the containers and waits to be lifted. What is the minimum power of the motor for this lift to work?

(4 points)1. Series 25. Year - 5. recoil

figure

When shooting from a gun the resulting backward impact causes the bullet to shoot out in a different direction than was originally meant. What is the angle difference between these two directions? Assume that hand muscles compensate for any influence that gravity can have. Also assume that the gun is rotating only around some point in a wrist. You know the moment of inertia of the gun-hand system (with respect to the previously mentioned point) as well as the mass of the projectile, its speed when leaving the gun and the dimensions described in the picture. After you solve this problem qualitatively make a quantitative guess of the necessary quantities and find a numerical value for the angle.

Unknown shooter

6. Series 24. Year - 1. warm up

figure
crooked table

Small ball is moving along a horizontal table from one end to the other with some initial velocity. In which case is the time required for the ball to traverse the table the shortest? Explain your choice. - Table has a concave bow. - Table has a convex bow. - Table is flat. - The curvature of the table does not matter.

broken bridge

Small valley of width $L$ is bridged using a board that is broken in the middle. It is however not entirely broken and still holds together so that its shape resembles that of a graph of absolute value. We place a small ball at one end and let it go. What is the appropriate depth of this bridge so that the time required for the ball to get to the other side is the shortest? Assume the ball does not lose energy while rolling over the bridge. You may need to know that the function $f(x) = x+1/x$ has a minimum at $x=1$.

5. Series 24. Year - 3. heavy chain

A chain of mass $m$ and length $l$ is hanging right above a scale. Initially it is at rest but then it starts falling. How does the scale's reading depend on the length $x$ of the chain that is already laying on the scale? Assume that the size of single chain cells is negligible.

Karel

4. Series 24. Year - 1. Warm-Up

 

  • Strings.

Using dimensional analysis determine the dependance of the frequency of oscillatons of a string if you know that it depends on its length $l$, on the tension $F$ in the string and on its linear density $ρ_{l}$.

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  • Downward.

You have a dumbell which consists of a short rod and two heavy discs. You wrap a string around the rod and let the dumbell fall while holding the string. What is the velocity of the dumbell? The discs have mass $M$ and radius $R$. The radius of the rod is $r$ and you can neglect its mass.

Karel, Jakub

3. Series 24. Year - 2. Parking strategy

You would like to park your car in a gap between two other cars on one side of a road. These cars are parked parallel to the road. Your car has legth $L$, width $d$ and the separation of wheels is $l$. The maximum angle the wheels can rotate by is $α$. What is the minimum length of this gap so that you are still able to park there? Assume both situations when you want to park going only forward or only backward. What is the ideal parking strategy?

Mára while watching the film.

2. Series 24. Year - 4. Think or pay

Suppose you are riding a bicycle and want to stop. What are the conditions so that the front wheel is completely blocked and sliding but you are not flying over your handle-bars? What effect does it have on your preceding result if you also use the break on the back wheel?

eee

1. Series 24. Year - 3. magical top

figure

Jakub has a spinning top with a spiral drawn on its upper side. He lets the top spin and watch it from above. What does he see and why?

Na dětství zavzpomínal Jakub

1. Series 24. Year - P. to be, or not to be

Two travelers, one fat and the other skinny, are arguing who would have better chances of surviving in extreme conditions. Tell them who will live longer in the following environments. Hot(50 °C), cold(-1 °C), after a boat accident in the Mediterranean sea, inside a hurricane, inside a heavy snow storm and in the middle of earthquake inside a city. Except of their body fat they are exactly the same. They even wear the same clothes and they do not carry anything else. Be original and remember that details matter.

Ve známém televizním pořadu viděl Honza P.

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