Problem Statement of Series 2, Year 27

The chocolate bar Twix is 32 % coating. Assume that it has a shape of a cylinder with a radius of 10 mm. Neglect the coating of the base. How thick is the coating?

Bonus: Think of a better model of said bar.

We have a wooden sphere at a height of $h=1\mathrm{m}$ above the surface of the Earth which has a perimeter of $R_Z=6378\mathrm{km}$ and a weight of $M_Z=5.97\cdot {10}^{24}\mathrm{kg}$. The sphere has a perimeter of $r=1\mathrm{cm}$ and is made of a wood which has the density of $\rho =550\mathrm{kg}\cdot {\mathrm{m}}^{-3}$. Assume that the Earth has an electric charge of $Q=5C$. What is the charge $q$ that the sphere has to have float above the surface of the Earth? How does this result depend on the height $h?$

We have a container of a constant cross section, which contains an ideal gas and a piston at a height of $h$. First we compress the air quickly (practically adiabatically) by moving the piston to a height of $h⁄2$, we hold it there until thermal equilibrium with its surroundings is reached, and then we let it go. To what height will the piton rise immediately? What is the height that it will reach after a very long time? Draw a $pV$ diagram.

It is known that the Moon when it is full has the apparent magnitude of approximately -12 mag and the Sun during the day has the apparent magnitude of -27 mag. Try to figure out what is the apparent magnitude of the Moon directly before a solar eclipse, if you know that the albedo of the Earth is approximately 0.36 and the albedo of the Moon 0.12. Presume that light after reflection disperes the same way on the surface of both the Moon and Earth.

A truncated cone that is the upside down (the hole is open downwards) may be held in the air by a stream of water which originates from the ground with a constant mass flowrate and an intial velocity $v_0$. At what height above the surface of the Earth will the cone levitate ?

Bonus: Explore the stability of the cone.

Estimate how much nuclear fuel get used by an atomic powerplant to generate 1 MWh of electrical energy that people use at home. Compare it with the usage offuel in a thermal powerplant. Don't forget to think about all posible ways that energy gets lost.

Bonus: Include the energy that is required to transport the fuel into your solution.

Let us have an inclined plane on which we place a ball and we give it kinetic energy so that it will begin rolling upwards without slipping. Measure the relationship between the velocity of the ball and time and determine the loss of energy as a function of time. The inclined plane should have an angle of at least $\alpha 10°$ with the horizontal. Do not forget to describe the parameters of your ball.

• What are the physical dimensions of action? (What are its units?) Does it have the same unit as one of fundamental constants from the first question in the previous part of the series? Which one?

• $FromNielsBohr$ – Assume the motion of a point mass on a circle with the centripetal force of

$$F_\;\mathrm{d} = m a_\mathrm{d} = \frac{\alpha}{r^2}\,,$$

where $ris$ the radius of a circle and $\alpha$ is some constant. Then

• Calculate the reducted action $S_0$ for one revolution as a function of its radius $r$.
• Determine the values of $r_n$, for which the value of $S_0$ is merely the constant from the sub-task a) multiplied by a natural number.
• The total energy of the point mass is $E=T+V$. For this force it istrue that $V=-\alpha ⁄r$. Express the energy $E_n$ of the particles depending on the radii $r_n$ using said constants.

Tip Youshould have encountered radial motion in your high-school education and also the relationships between displacement, velocity and acceleration. Use them and then the integration of action along the circumference of the circle with a constant $r$ shall become easier (constant quanties can be easily factored out of the integral). Don't forget that the path integral of „nothing“ is merely the length of the integrated path.

• The last sub-problem may seem complicated but it is merely a excercise in differentiating and integrating simple functions. You should be able to do it nly with standard table integrals and derivatives. Show that the full action $S$ for a free particle moving from the point [ 0$;0]$ to the point [ 2$;1]is$ for the case of linear motion (first case) minimal. In other words that it is bigger in the two other cases

$$\mathbf{y}(t)&=\left(2t,t\right) \,,\\mathbf{y}(t)&=\left(1-\cos{(\pi t)} \frac{1}{\pi}\sin{2\pi t}, t\right) \,,\\mathbf{y}(t)&=\left(2t, \frac{\;\mathrm{e}^t-1 t^2(t-1)}{\mathrm{e}-1}\right) \,,$$

where e is the Euler number. Tip First find the derivative of $\mathbf{\text{y}}(t)$, put it into the equation for action and integrate.