Assignment of Series 1 of Year 32

How many balloons of volume $V=10 \mathrm{\ell }$ filled with helium of density $\rho _{\scriptscriptstyle \rm He} = 0{,}179 \mathrm{kg\cdot m^{-3}}$ are needed to lift Filip, whose mass is $m_{\scriptscriptstyle \rm F} =80 \mathrm{kg}$, and keep him afloat in air of density $\rho _{v} =1{,}205 \mathrm{kg\cdot m^{-3}}$? How many would be necessary to lift Danka, who weighs $m_{\scriptscriptstyle \rm D} =50 \mathrm{kg}$? Neglect the mass of the empty balloons.

Jachym was launching fireworks. We can imagine it as a beacon, that is at some point shot straight up with velocity $v$, and explodes after a certain delay. Jachym was standing a distance $x$ from the launch site when he heard the launch of the fireworks. After a delay $t_1$ he saw the explosion and after another delay $t_2$ he heard the explosion. Calculate the velocity $v$.

We have 8 point charges (each of magnitude $q$) located on the vertices of a cube. Find out the value of a point charge $q_0$ that needs to be placed in the middle of the cube, so that all charges remain in balance. Is this equilibrium stable?

When James Bond let go of agent 006 Alec Treveljan from the top of the Arecibo radiotelescope in the final scene of the film Golden Eye, the falling agent started screaming with a frequency $f$. How does the frequency agent 007 hears at the top of the telescope change as a function of time. Neglect air resistance.

Hint: Ask Mr. Doppler

a) Determine the resistance between points A and B of the infinite grid in the picture. The point A is directly connected to two resistors with resistances $R_a$ and $R_b$. Each of these resistors is connected to two more resistors with $R_a$ and $R_b$ etc.

b) Replace all the resistors with capacitors of capacitances $C_a$ and $C_b$. What is the total capacitance of the circuit?

Some nebulas constituted of a gas from stars, e. g. Bumerang, have lower temperature than the Cosmic Microwave Background (CMB), hence are technically colder than space. How is this possible? Try to determine a condition for a gas ejected by a hot star to cool down below the temperature of the CMB.

Measure the length of one day. However, there is a limitation: one continuous measurement can't take longer than one hour. For the sake of statistical accuracy, though, do repeat your measurements multiple times.

Before we dive into the art of analytical mechanics, we should brush up on classical mechanics on the following series of problems.

1. A homogenous marble with a very small radius sits on top of a crystal sphere. After being granted an arbitrarily small speed, the marble starts rolling down the sphere without slipping. Where will the marble separate and fall of the sphere?
2. Instead of the sphere from the previous problem, the marble now sits on a crystal paraboloid given by the equation $y = c - ax^2$. Again, where will the marble separate from the paraboloid?
3. A cyclist going at the speed $v$ takes a sharp turn to a road perpendicular to his original direction. During the turn, he traces out a part of a circle with radius $r$. How much does the cyclist have to lean into the turn? You may neglect the moment of inertia of the wheels and approximate the cyclist as a mass point.
   Bonus: Do not neglect the moment of intertia of the wheels.