## Solutions for theoretical problems in Physics## December, 2001 |

In this page only the sketch of the solutions are published; in some cases only the final results. To achieve the maximum score in the competition more detailed solutions needed.

**P.** **3478.** Estimate how much work you do
while you reach a proper press-up position from lying. (3 points)

**Solution.** *W\(\displaystyle approx\)*100-200 J.

**P.** **3479.** Hydrogen gas is kept streaming
through a tube while it is cooled by water surrounding the tube. The
temperature of the incoming gas is 60 C^{o} and that of the
outgoing gas is 30 C^{o}. The capacity of the cooler tank is
60 liter and the water in it is completely changed in every hour. The
initial temperature of the cooling water is 10 C^{o} and it is
20 C^{o} when it is changed. What is the amount of the gas in
kgs that goes through the tube in one hour? (4 points)

**Solution.**

*m*_{hydrogen}=5.9 kg.

**P.** **3480.** From a 5000 km radius planet
a rocket is launched at the first cosmic velocity in a direction at a
60^{o} angle to the vertical direction. What is the maximum
height the rocket can reach? (5 points)

**Solution.** *h*=*R*cos\(\displaystyle alpha\)=2500 km.

**P.** **3481.** What is the relation between the
temperature and the swing period of a physical pendulum made of a
homogeneous material? (4 points)

**Solution.**

\(\displaystyle {\Delta T\over T}={1\over2}\alpha\Delta t.\)

**P.** **3482.** On the middle one of three thin
concentric spherical metal shells of radius *R* there is *Q*
electric charge. The inner spherical shell of radius
and the outer spherical shell of radius
are earthed.

*a*) What is the amount of the electric charge on the earthed
spherical shells?

*b*) Plot the electric field strength versus the distance from
the centre. (5 points)

**Solution.**

\(\displaystyle E(r)=\cases{-k{\displaystyle1\over\displaystyle4}{\displaystyle
Q\over\displaystyle r^2},{\rm if\ }{1\over2}R

**P.** **3483.** In the connection shown in the
*figure* the switch K is open and the capacitor is
uncharged. Then we close the switch and let the capacitor charge up to
the maximum and open the switch again. Determine the values indicated
by the ammeter

*a*) directly after closing the switch;

*b*) a long time after the switch was closed;

*c*) directly after reopening the switch;

*U*_{0}=30 V, *R*_{1}=10 k,
*R*_{2}=5 k\(\displaystyle Omega\).) (4 points)

**Solution.** *a*) After closing the switch *I*0.

*b*) After a long time

\(\displaystyle I={U_0\over R_1+R_2}=2~\rm mA.\)

*c*) After reopening the switch *I*2 mA.

**P. 3484.** What should be the capacity of the adjustable
capacitor shown in the *figure* so that the voltmeter indicates
the highest voltage? (5 points)

**Solution.** The voltage of the capacitor is

\(\displaystyle U_C={I\over\omega C}={U_0\over\sqrt{R^2\omega^2C^2+(1-\omega^2LC)^2}}.\)

This function has a maximum at

\(\displaystyle C={1\over L\omega^2+R^2/L}.\)

**P.** **3485.** The Sun attracts the Earth with a
lot greater force then the Moon does.

*a*) What is the ratio of the two forces?

*b*) What is the explanation for the fact that despite this
the tidal effect of the Moon is a few times bigger than that of the
Sun. (5 points)

**Solution.** *a*)

\(\displaystyle {F_{\rm Sun}\over F_{\rm Moon}}={M_{\rm Sun}\over M_{\rm Moon}}\cdot\left({r_{\rm Moon}\over r_{\rm Earth}}\right)^2\approx170.\)

*b*) The tidal effect is proportional to the inhomogeneity
(derivative) of the gravitational force:

\(\displaystyle F_{\rm tidal}\propto{dF(r)\over dr}\propto{M\over r^3}.\)

So

\(\displaystyle {F_{\rm Sun}^{\rm tidal}\over F_{\rm Moon}^{\rm tidal}}={M_{\rm Sun}\over M_{\rm Moon}}\cdot\left({r_{\rm Moon}\over r_{\rm Earth}}\right)^3\approx0.5.\)

**P.** **3486.** The binding energy of the
^{14}N nucleus is 16.19 pJ and the ^{14}C nucleus
is 16.37 pJ. Which nucleus is the decay product of the other, and
why? (5 points)

**The solution will appear at the end of April.**

**P.** **3487.** Werner Heisenberg, the Nobel Prize
winner German theoretical physicist, was born a hundred years ago
(5^{th} December 1901). With the help of the uncertainty
principle introduced by him estimate the uncertainty of the velocity
of a carbon atom and an electron `sitting' on the point of a pine
needle! (4 points)

**Solution.** \(\displaystyle Delta\)*v*_{electron}\(\displaystyle approx\)1 m/s, *v*_{carbon}\(\displaystyle approx\)10^{-5} m/s.