Serial of year 22

• What had to be a conditions for Galileo to fail its experiment? Leaning tower in Pisa is $h=55\;\mathrm{m}$ high. Assume that both balls have the same diameter $R=8\;\mathrm{cm}$ and that one is made from lead of density $ρ=11300\;\mathrm{kg}\cdot \mathrm{m}^{-3}$. What density must be the second ball to achieve the difference in time of impact to be bigger than $ΔT=0,3s?$

• What is precision of original Eötvös measurement of equality of gravitational and inertial mass for neutrons and protons, if in wood neutrons make 50 percent of mass and in platinum 60 percent of mass? Neglect mass of electrons and binding energy.

• Verify the assumption, that in Budapest is $g_{s}′$ negligible comparing to $g$.

• What shape of interference pattern would you expect in following two configurations? Find the equations of curves of maximum intensity and try to plot some of them.

• Show, how the experiment should end, if the light behaves according to Newton assumptions. Do consider different angle of incidence at different places on the screen.

• Using quantum-mechanical description find resolution, which should have get Jöhnsson by using 4 slits (4 parallel orifices in distance $b$ between neighbouring slits. Plot representative part of the graph and comment advantages of higher number of orifices.

* Imagine a strong laser at wavelength 400 nm, and shine it at the Moon. On its surface the light will reflect and come back. Assuming circular orifice of diameter of 1 cm through which the beam is going, what will be the diameter of the reflection on the Earth? Hint: It will be much more, than 1 cm.

• In this task assume, that the aether really exists and predict, what will happen, if Mr. Michelson would make its measurement by other means: one arm would be 5 meters long and other 10 meters long. Such apparatus would create some interference pattern. Then he would rotate whole experiment by 90$°$, so both arms changed its positions. During rotating the experiment, we would see changes in interference patterns (assume rotating doubleslit). How would the interference patterns move at above rotation? How long would have to be the longer arm to inverse the interference fringes (e.g. minima would become maxima)?

• In the following task again assume existence of aether and that a body moving in aether is pulling it completely with to body, so the relative speed of aether to the body is zero. What would be then phase shift between two beams in the system in above figure? The light is splitted at semitransparent mirror into two beams and continues at perfectly rectangular path back to the semitransparent mirror, where it reaches screen at which interference fringes are observed. On the way are both beams three times reflected by a mirror and are going through the cylinder of length $L$, filled with water. Whole system is moving relatively to aether at speed $v$ to the right (do not forgot, that the cylinder is not moving relatively to aether!).

• Foucault pendulum has drawn into a sand at two different demonstrations two different diagrams, as on the picture. Find out, what caused different shape and how long the pendulum was to create such diagrams on the floor of Paris cathedral. How many tips do the stars have in reality?

• What shape will have the water level in barrel standing still at horizontal plane?

• Show, that equation

$$δf=f_{+}-f_{-}=4**ω**\cdot \textbf{S}/(λ_{0}P)$$ for frequency difference (frequency of beats) of two contra-propagating light beams in laser gyroscope, is valid for any shape of gyroscope, not only circle.

• The second method to measure specific electron charge used by J. J. Thomson is observation of deflection of cathode ray by electric field. Assume apparatus as on the figure. How depends deflection of the beam observed on the screen on the right on the electrical voltage, speed and geometrical dimensions of apparatus?

• The one of problems which was J. J. Thomson facing during measurement of specific electron charge was following: After the beam entered magnetic field, the beam has spread into a bigger area (see figure). The dispersion is causing some error in measuring position of electrons (Thomson mentioned up to 20%). How can be this dispersion explained? How can be this inaccuracy improved (this is for bonus point)?

• By using the data from table calculate charge of electron in case, that the oil had density of 920 kg\cdot m^{ − 3}, air density was 1,2 kg\cdot m^{ − 3} and viscosity 17,1 \cdot 10^{−7} Pa\cdot s. Used electric field was 250 kV\cdot m^{ − 1}.

• Decide if the stability (e.g. dimensions) of Saturn model depends on atomic number $Z$.

• Change equation (12) for probability of scatter of $α-particle$ at high angle $φ$ in such way, to get more practical equation for probability of impact per unit area on scintillator. and show, how this can be used to get material of target. Further estimate how the equation would change when not considering the central charge $Ze$ but $Z$ spread elementary charges $e$ as is for example in Lenard model.

• In 1896 astronomer E. C. Pickering found in the spectrum of star $ζ$ Puppis lines, which fulfilled condition (7) for $n=2$ and $m=2,5;$ 3; 3,5; 4; 4,5;…,

e.g. also for half integers. Explain this inconsistency of Bohr model.

• ( Bonus: Find a dependence similar to equation (11) for Thompson pudding model and comment on differences. Or try to modify it in such way, that it considers all atoms in thin aluminium foil. Just play a little bit.