# Series 3, Year 24

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### 1. Warm-Up

*Dr. Nec.*

There are two ways to measure the amount of wood in a pile of trees. Either as the volume of pure wood in the pile or as the volume of wood together with the empty spaces in the pile. Find the conversion factor between these two units assuming the trees are cylinders of radius $r$ that are layed one on top of the others.

*Bubbles.*

A spherical cap of radius $r$ is made by blowing air into a circular surface of soap water. Estimate the velocity of air molecules hitting the surface?

Jakub

### 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.

### 3. Aleš, the Drug Addict

Aleš stores toluen in a cylindrical bottle. When he left the room 90% of the bottle was occupied by toluen. When he returned after the weekend he noticed that the level of toluen was lower. He obviously accused his roommate from stealing. Then he realized that over the weekend the temperature in the room had risen by 20$\celsius$.
**Help:** Aleš solve this mystery. Was his roommate really stealing or could the temperature rise be responsible? You can use data from http://en.wikipedia.org/wiki/Toluene (data page).

Mára was filling a bottle with toluene.

### 4. Straws

Two straws are connected so that the resulting object is of the shape of letter V. This object is then supported on its sides so that it can swing. Determine the stability conditions (the object should not slide) and also the period of oscillations of this system. The radius of the straws is $r$.

Jakub.

### P. Water, water and water

There are many interesting properties of water that other liquids do not possess. Some of them are listed at http://www.btinternet.com/ martin.chaplin/anmlies.html. Think of some consequences of these anomalies for the life on Earth, humans and technology.

Mára was listening to Meteor.

### E. Paper

Experimentally determine the dependance of transparency of a paper on the incidence angle of light.

Jakub.