Series 3, Year 15

A giant and a dwarf are tugging of war. The rope is reeled around a tree. The rope and the tree are absolutely unbreakable.

The bad giant is exactly 666 times stronger than the dwarf. How many times must the rope be reeled around the tree if we don't want the giant to win? Estimate the coefficient of friction.

Estimate the total kinetic energy of a pair dancing Viennese waltz.

Describe the shape of an icicle growing from a rotating wheel of a ski-lift. The angle between the wheel and the horizontal plane is denoted $\alpha$, angular velocity of the wheel is $\omega $ and the icicle starts growing at a distance $r$ from the the axis.

The Global Positioning System (GPS) is based on a rather simple principle. Satellites on 12-hour orbits emit perfectly synchronized signals then detected by a receiver. The receiver cannot carry perfect clock and therefore can detect only differences of distances to the satellites (i.e. it cannot measure the distance, but only the difference between two distances). Four satellites are enough to calculate the position.

Explain, why is the accuracy of GPS significantly better in the horizontal direction than in the vertical direction.

Round up two magnets and a thin iron plate. Place the magnets against each other, put the plate between them and check the attracting force. Than turn one of the magnets and repeat the experiment. Also check the attractive and repulsive forces with the plate removed.

You will probably find something strange about the experiments. Describe the behaviour of the magnets and explain it.

Measure the coefficient of reflexivity of aluminium foil in the visible light. Suggest an appropriate method. Do not forget to describe the side of the foil you measure.

In this problem we analyse and interpret measurements made in 1994 on radio wave emition from a source consisting multiple bodies within our galaxy. The distance to the central celestial body from Earth is estimated to be $R = 3,86.10^{20}$ m. The angular velocities of two objects ejected from the centre in opposite directions were measuredyo be: $\omega _{1} = 9,73.10^{-13} rad.s^{-1}$ and $\omega _{2} = 4,42.10^{-13} rad.s^{-1}$. We calculate the transverse velocities: $v_{1} = R\omega _{1} =3,76.10^{8} m.s^{-1}$ and $v_{2} = R\omega _{2} = 1,71.10^{8} m.s^{-1}$. The first object is faster than light! How is it possible?

Let's consider an object moving with velocity $v$. The angle between the velocity vector and the direction to the observer is $\varphi$. The distance to the observer is denoted $R$. Calculate the angular velocity as seen by the observer. Can $Rω$ be greater than the speed of light? Using your results calculate the real velocities of the two objects. Assume that the velocities are equal.