Series 2, Year 24

• Jakub's breakfast

Every morning Jakub enjoys his favourite cereals which he pours into a bowl of milk. Assume the bowl is of circulur frustum shape with upper and lower radius $R$ and $r$ respectively ($R$ \geq r$)$, that the cereals are little solid spheres and that before he puts the cereals into the bowl there is milk of height $h$. What is the maximum amount of cereals he can fit into the bowl? You also know that the fraction of volume the cereals occupy inside a fully filled box is $\kappa$.

• Magnetic monopole

Let's have a metal plate magnetized in such a way that the upper and lower sides are the north and south poles respectively. We use these plates to create two semispheres with the outer side being the north pole. Now, if we glue these two semispheres together, we effectively get a magnetic monopole, which, as we know, can not exist in our world. Where did we go wrong?

The interaction of two atoms of an inert gas can be described using the so called Lennard-Jones potenial $U(R) = 4\epsilon((\sigma/R)^12 - (\sigma/R)^6)$. Assume the motion is one-dimensional and determine the equilibrium position without using the tools of calculus. The meaning of the constants $\sigma$ and $\epsilon$ will be explained in the published solution.

While enjoying his daily coffee, Lukáš decided to tune up his percolator a bit. He placed a bent tube with a short wire wrapped around it to the bottom of the main vessel (see the picture). The wire was placed height $d$ above the bottom and the vessel's water level was in a height $h$. Parameters of the tube were chosen in such a way that the water vapor created by boiling water close to the wire pushed the water above it. What is the power we need to provide to the wire in order to see water coming from the tube in a height $l?$

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?

If you inhale some helium your voice changes so that you speak like a smurf. Hydrogen has the same effect (watch out smokers!) but it is also possible to obtain the voice of Darth Vader in this way. The most famous substance for this is sulfur hexafluoride. What causes the voice to change? Make also a quantitative guess.

Most of you have probably heard about the Young's double slit experiment. Have you, however, ever tried to reproduce this experiment and see the interference patterns for yourselves? There are also mechanical analogies to this experiment. For example you can observe the interference of two waves in water or two sound waves. Choose one or more of these experiments and measure the interference pattern. Then you can calculate the wave length and the speed of wave propagation. Photos of your apparatus will be welcomed!