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Railway has a shape of an arc of radius $R$ and its width is $D$. Train's center of mass is at height of $H$ meters. How fast can a train go on this railway if, no matter where it stops, it never falls? Under what conditions is this maximum speed unbounded? Note Neglect forces cars act on each other and also assume the width of a car is much smaller than the radius $R$.

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

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?

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.

crooked table

broken bridge

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.

• 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.

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?

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?

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?