# Physics Learning Center

## Physics Olympiads Series Problem 4

Problem 4
A hydrogen atom in the ground state, moving with velocity , collides with another hydrogen atom in the ground state at rest. Using the Bohr model find the smallest velocity v0 of the atom below which the collision must be elastic. At velocity v0 the collision may be inelastic and the colliding atoms may emit electromagnetic radiation. Estimate the difference of frequencies of the radiation emitted in the direction of the initial velocity of the hydrogen atom and in the opposite direction as a fraction (expressed in percents) of their arithmetic mean value.
Data:
1(ionization energy of hydrogen atom):

2(mass of hydrogen atom):

( m - mass of electron; e - electric charge of electron; ħ - Planck constant; numerical values of these quantities are not necessary.)
(Taken from: Problems of the 7th International Physics Olympiad (Warsaw, 1974)-Theoretical Problems#1 )

## Physics Olympiads Series Problem 3

Problem 3
Inside a thin-walled metal sphere with radius R = 20 cm there is a metal ball with the radius r = 10 cm which has a common centre with the sphere. The ball is connected with a very long wire to the Earth via an opening in the sphere (Fig. 3). A charge Q = 10-8 C is placed onto the outside sphere. Calculate the potential of this sphere, electrical capacity of the obtained system of conducting bodies and draw out an equivalent electric scheme.

Fig.3
(Taken from: Problems of the IV International Olympiad, Moscow, 1970-Theoretical Problems#3 )

## Physics Olympiads Series Problem 2

Problem 2
Consider an infinite network consisting of resistors (resistance of each of them is r) shown in Fig. 3. Find the resultant resistance RAB between points A and B.

Fig.3
(Taken from: I IPhO (Warsaw 1967-Theoretical problems-Problem 2)

## IPhO VIII 1975 Güstrow Problem 1

This sample is taken from Problems of the 8th International Physics Olympiad (Güstrow, GDR, 1975), problem 1.

Problem 1
Theoretical problem 1: “Rotating rod”
A rod revolves with a constant angular velocity ω around a vertical axis A. The rod includes a fixed angle of π/2-α with the axis. A body of mass m can glide along the rod. The coefficient of friction is µ = tan β. The angle β is called „friction angle“.
a) Determine the angles α under which the body remains at rest and under which the body is in motion if the rod is not rotating (i.e. ω = 0).
b) The rod rotates with constant angular velocity ω > 0. The angle α does not change during rotation. Find the condition for the body to remain at rest relative to the rod.

You can use the following relations:
sin (α ± β) = sin α ⋅ cos β ± cos α ⋅ sin β
cos (α ± β) = cos α ⋅ cos β ± sin α ⋅ sin β

## IPhO VII 1974 Warsaw Problem 3

This sample is taken from Problems of the 7th International Physics Olympiad (Warsaw, Poland held in 1974), problem 3.

Problem 3
A scientific expedition stayed on an uninhabited island. The members of the expedition had had some sources of energy, but after some time these sources exhausted. Then they decided to construct an alternative energy source. Unfortunately, the island was very quiet: there were no winds, clouds uniformly covered the sky, the air pressure was constant and the temperatures of air and water in the sea were the same during day and night. Fortunately, they found a source of chemically neutral gas outgoing very slowly from a cavity. The pressure and temperature of the gas are exactly the same as the pressure and temperature of the atmosphere.

The expedition had, however, certain membranes in its equipment. One of them was ideally transparent for gas and ideally non-transparent for air. Another one had an opposite property: it was ideally transparent for air and ideally non-transparent for gas. The members of the expedition had materials and tools that allowed them to make different mechanical devices such as cylinders with pistons, valves etc. They decided to construct an engine by using the gas from the cavity.
Show that there is no theoretical limit on the power of an ideal engine that uses the gas and the membranes considered above.

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