Problem Statement of Series 2, Year 37

When working out, we often come across workout machines that contain pulleys. Consider the machine in the following figure. What force must be applied on the rope if the velocity of the end of the rope at point A is $v=0.4\mathrm{m}\cdot {\mathrm{s}}^{-1}$ and its direction is downwards? Each pulley has a radius $r=15\mathrm{cm}$ and mass $m=15\mathrm{kg}$. A weight of mass $M=25\mathrm{kg}$ hangs over the free pulley.

It is said that if you want to inflate a car's tires, you should do it when they are cold. Therefore, Jarda drove to a petrol station with a compressor, bought a hot dog, and waited for the tires to cool down. Curious, he measured the tire pressure before and after his snack. It had dropped from $2.7\mathrm{bar}$ to $2.5\mathrm{bar}$. He wondered whether the tire pressure could be determined by the height of the car's body above the road. How much did the body of Jarda's car approach the ground due to the decrease in the tire temperature? The weight of the car is $1.3\mathrm{t}$. The outer radius of the tires is $32\mathrm{cm}$, the inner radius is $22\mathrm{cm}$, and their width is $21\mathrm{cm}$. Assume that the tires deform due to the car's weight only on the underside where they touch the ground.

Consider a planet with the same total mass and radius as the Earth. How much uranium ${\phantom{A}}_{\phantom{}}^{\phantom{238}}{\phantom{A}}_{\phantom{2}}^{\phantom{2}238}\mathrm{U}$ would it have to contain, so that its surface temperature is $15{}^{\circ }\mathrm{C}$, assuming it is not lit by any nearby star.

Lego wanted to take a break from a problem in his thesis, where a quantum heat machine behaved like a perpetual motion machine. Thus, he came up with a perpetual motion machine in classical physics using the following reasoning. Somewhere in a pit (it doesn't even have to be very deep) we evaporate water (to do this we consume some latent heat). The water rises as vapor upwards, where we condense it again (releasing the latent heat). But the water now has a higher gravitational energy! Where did this energy come from? Or should Lego run to the patent office to go down in history as the inventor of the perpetual motion machine? Support your claims with calculations.

Imagine a ferry in the shape of a rectangular cuboid with a weight $M$, length $L$, width $W$, and height $H\ll L$ from the keel to the deck. After docking at the pier, passengers gradually exit through the back of the deck so that the empty front part of the deck becomes larger and the area density of people on the filled part does not change in a different way. Find the maximum weight of passengers the ferry can carry so that no part of the deck is below the surface when people disembark. Consider that the ship is stable in the transverse direction and that people get off slowly.

Which factors influence the height of mountains on different planets? Make an attempt at a quantitative estimate. You can consider the highest mountains on the Earth, Mars, and other known planets.

Measure the illumination intensity of light passing through a cola as a function of the drink's thickness. Determine the absorption coefficient by curve fitting the measured data.

1. Measure your elbow in inches. Use only your body parts for the measurement.
2. In ancient times, the first attempt to determine the distance of the Earth from the Sun was to measure the angular distance of the Moon from the Sun when the Moon was in the first quarter – the interface of light and darkness was direct. Determine the magnitude of this angle and compare it with the angular size of the Earth as seen from the Moon.
3. A laser distance meter using a $\mathrm{He}$-$\mathrm{Ne}$ laser shows the distance exactly $100\mathrm{m}$ under standard conditions $(20{}^{\circ }\mathrm{C},100\mathrm{kPa})$. How will this value differ when the following changes:
   • temperature by $30{}^{\circ }\mathrm{C}$
   • pressure by $10\mathrm{kPa}$
   • a green laser with a wavelength of $532\mathrm{nm}$ will be used instead
   • no conversion between group and phase velocity
4. State at least $4$ different ways of measuring the velocity of vehicles. Explain which physical principles are used to determine the velocity and which type of velocity it is.