## Solutions for theoretical problems in Physics## November, 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.** **3466.** Looking into a convex mirror our
nose seems to be disproportionately larger than our ears. Why? (3
points)

**Solution.** The distance between our ears and the mirror is
larger than the distance between our nose and the mirror.

**P.** **3467.** Cycling without rest between two
towns the journey takes 6 hours. What is the distance between the two
towns if our average speed is 16 km/h on level surface,
12 km/h uphill and 24 km/h downhill? (3 points)

**Solution.** *d*=48 km.

**P.** **3468.** In a heat insulated vessel there
is tap water of 15 ^{o}C. We place in it ice cubes of
-15 ^{o}C taken out of the deep-freezer. May there be no
change of state? When does only a part of the ice melt? In which case
does all of the ice melt? In which case does the water freeze? (4
points)

**Solution.** Let us denote the mass ratio of the ice and the
water with *r*!

- If *r*<0,17, all of the ice melt and the temperature will
be positive.

- If 0,17<*r*<2, only a part of the ice melt and the
temperature will be 0 ^{o}C.

- If
2<*r*<12,7, a part of the water freeze and the temperature
will be 0 ^{o}C.

- If 12,7<*r*, all of the water freeze and the temperature
will be negative.

**P.** **3469.** The T shaped rigid body shown in
the *figure* consists of three uniform rods, and can rotate round
the horizontal axis *t* perpendicular to its plane. At the end of
the right side rod and at the point dividing it in a 2:3 ratio there
are ropes fixed and each of the ropes hold horizontal rods propped up
by wedges. The wedge holding the upper rod divides the lower rod in
3:2 ratio. The mass of the upper rod is *m*_{1}, that of
the lower one is *m*_{2}, and the mass of the body lying
on the upper rod is *m*_{3}. What is the required mass of
the body that is to be hung on the end of the left hand side rod so as
to keep the balance. The data: *m*_{1}=2.5 kg,
*m*_{2}=6.0 kg, *m*_{3}=7.5 kg.
(5 points)

**Solution.** \(\displaystyle m={2\over5}(m_3+m_1)+{1\over2}m_2=7.0\) kg. The result does
not depend on the position of the body *m*_{3}.

**P.** **3470.** A carriage is joined to the end of
a slope as can be seen in the *figure*. A body released from the
slope from height *h* slides as far as the middle of the
carriage. (The friction between the body and the slope and the mass of
the wheels of the carriage are negligible.) At what height should the
body be released so that it could stop just at the end of the
carriage?
(4 points)

**Solution.**

\(\displaystyle s={1\over\mu}{M\over m+M}h\propto h,\)

so the height should be 2*h* in the second case.

**P.** **3471.** We make a spiral of 0.6 m
radius and 30^{o} slope angle of a thin but sufficiently stiff
wire and fix it in a way that its axis is vertical. Then we thread a
pierced pearl on the wire and at a given moment we let it slide
down. What speed does it gather if the friction coefficient is 0.5?
(5 points)

**Solution.**

\(\displaystyle v=\sqrt{{rg\over\mu\cos\alpha}\sqrt{{\rm tg}^2\alpha-\mu^2}}\approx2~{\rm m\over\rm s}.\)

**P.** **3472.** We fix a block to a
spring. Hanging the spring at its end the block oscillates with a
period of 1 second. Then we put the block on a horizontal table and
hurl it towards a wall so that the spring is positioned in between as
a bumper. Colliding with the wall the spring contracts then hurls the
body back. The block stops exactly when the spring gets its neutral
size back. How much time elapsed between the spring touching the wall
and the the block stopping? (4 points)

** Solution.**

\(\displaystyle \Delta t=T\left({3\over4}-{{\rm arcsin}(1/3)\over2\pi}\right)=0.7T=0.7~\rm s.\)

**P.** **3473.** What are the mass percentages in a
mixture of hydrogen and helium gases where at isobar dilation 70 % of
the absorbed heat from the surrounding environment is converted to
internal energy. (4 points)

**Solution.** The mass percentage of helium is 28.6%.

**P.** **3474.** At the endpoints of a line segment
of a length of *d* there are two identical positive electric
charges. What is the ratio of the electric field strength and the
electric potential in a point characterized by an angle on the
Thales' circle drawn around the line segment. (4 points)

**Solution.**

\(\displaystyle {\vert E\vert\over U}={1\over d}{\sqrt{\sin^4\alpha+\cos^4\alpha}\over\sin\alpha\cos\alpha(\sin\alpha+\cos\alpha)}.\)

**P.** **3475.** On a bumper at the lower end of a
vertically positioned insulator bar shown in the *figure* there
is a pearl of the mass *m*=10^{-4} kg and a charge
of *Q*_{1}. Above at a *h*_{0}=20 cm
distance there is another pearl with the same mass *m* and
electric charge *Q*_{2} afloat. At a given moment we kick
the lower pearl and it starts upwards at a velocity of
*v*_{0}=2 m/s. How close can the lower pearl get to
the upper one at most? (The pearls can move along the bar without
friction.) (6 points)

**Solution.** Using the energy conservation in the centre of
mass system

\(\displaystyle s_{\rm min}={4gh_0^2\over v_0^2+4gh_0}\approx13~\rm cm.\)

**P.** **3476.** In a homogeneous magnetic field
there is an electron moving in a circular orbit. Can the magnetic
induction vector generated by the moving electron be greater in the
middle of the circle than the magnetic induction vector of the
homogeneous field? (5 points)

** Solution.**

The ratio of the generated magnetic field to the homogeneous field is

\(\displaystyle {B_g\over B_0}=\left[{e^2\over4\pi\varepsilon_0}{1\over mc^2}\right]\cdot{1\over R}={r_0\over R},\)

where *r*_{0}=2.8^{.}10^{-15} m
is the classical electron radius. In the region of
*R*<*r*_{0} the classical electrodynamics
collapses, consequently *B*_{g} can not be greater
than *B*_{0}.

**P.** **3477.** Using thin slats we construct a
triangle shaped frame with sides *a*, *b* and *c*. The
overall mass of the frame is *m*. What is the inertial momentum
of the frame related to an axis perpendicular to the triangle and
crossing the frame in its centre of mass? (5 points)

**Solution.**

\(\displaystyle \Theta_{\rm frame}={m\over12(a+b+c)}\left(a^3+b^3+c^3+3abc\right).\)