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IPhO 1968 Budapest Problem 2

This problem is taken from the second IphO Budapest, Hungary, Theoretical problems, in 1968

Problem 2
There are 300 cm3 toluene of 0°C temperature in a glass and 110 cm3 toluene of 100°C temperature in another glass. (The sum of the volumes is 410 cm°C .) Find the final volume after the two liquids are mixed. The coefficient of volume expansion of toluene β = 0.001(°C)−1 . Neglect the loss of heat.


Solution
If the volume at temperature t1 is V1, then the volume at temperature 0°C is V10 = V1/(1+βt1). In the same way if the volume at t2 temperature is V2, at 0°C we have V20 = V2/(1+βt2). Furthermore if the density of the liquid at 0°C is d, then the masses are m1 = V10 d and m2 = V20 d, respectively. After mixing the liquids the temperature is



The volumes at this temperature are V10 (1+βt2) and V20 (1+βt2). The sum of the volumes after mixing:



The sum of the volumes is constant. In our case it is 410 cm3. The result is valid for any number of quantities of toluene, as the mixing can be done successively adding always one more glass of liquid to the mixture.

Sources/Literatures :
-Waldemar Gorzkowski, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland
-Péter Vankó, Institute of Physics, Budapest University of Technical Engineering, Budapest, Hungary, Problems of the 2nd and 9th International Physics Olympiads (Budapest, Hungary, 1968 and 1976)
-The Olympic home page www.jyu.fi/ipho

 

IPhO 1968 Budapest Problem 1

This problem is taken from the second IPhO Budapest, Hungary, Theoretical problems, in 1968.

Problem 1
On an inclined plane of 30° a block, mass m2 = 4 kg, is joined by a light cord to a solid cylinder, mass m1 = 8 kg, radius r = 5 cm (Fig. 1). Find the acceleration if the bodies are released. The coefficient of friction between the block and the inclined plane μ = 0.2. Friction at the bearing and rolling friction are negligible.

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IPhO 1967 Problem 4

This problem sample below is taken from theoretical problems of first IphO that's conducted in Warsaw, Poland 1967, problem 4.

Problem 4
A closed vessel with volume V0 = 10 l contains dry air in the normal conditions (t0 = 0°C, p0 = 1 atm). In some moment 3 g of water were added to the vessel and the system was warmed up to t = 100°C. Find the pressure in the vessel. Discuss assumption you made to solve the problem.

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IPhO 1967 Problem 3

This problem sample is taken from theoretical problems of first IphO that's conducted in Warsaw, Poland in 1967.

Problem 3
Consider two identical homogeneous balls, A and B, with the same initial temperatures. One of them is at rest on a horizontal plane, while the second one hangs on a thread (Fig. 6). The same quantities of heat have been supplied to both balls. Are the final temperatures of the balls the same or not? Justify your answer. (All kinds of heat losses are negligible.)

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IPhO 1967 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)

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