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Physics Olympiad Samples

Physics Olympiad Samples PHYSICS SOME OF MIPT Olympiad SAMPLES'1999 The system of two weights with masses m1 and m2 are connected with weightless spring (Fig. 1.) The system is resting on the support S. The support S is quickly removed. Find the accelerations of each of the weights right after the support S is removed.
Figure 1

Closed from both ends cylinder contains two moving pistons. Pistons divide the cylinder as shown on Fig. 2. Pistons can move without friction in the cylinder and provide complete thermal insulation between the parts of the cylinder. Three parts of the cylinder are filled with different ideal gases at the same temperature. The temperature in the middle section of the cylinder is increased by 60 per cent (on the Kelvin scale). The other two sections of the cylinder are kept at the initial temperature. Determine the relative volumes of all sections after the new equilibrium will be established (that is, after the pistons stop moving).

Figure 2

The profile P in the form of two connected "hills" is put on the plane surface as shown on the Fig.3. The heights of the "hills" are 4H and H. A small body b is put on the top of the taller hill. The mass of this small body is six times less than the mass of the profile P. Initially the body b rests on the top of the higher hill, but a slight push to the right puts it in motion with negligibly small initial speed. The body b remains in full contact with P during all movement. Find the speed of the profile P at the moment when body b reaches the smaller hill's top. Neglect friction.
Figure 3
Condensers with capacities C, 2C, 3C and 4C are charged to the voltage V, 2V, 3V and 4V correspondingly (see Fig. 4). The circuit is closed. Find the voltage on all condensers in the equilibrium.
Figure 4
Consider an electrical scheme on the Fig.5. All parameters shown on the scheme are known. R is resistance. E is the battery's voltage, r is its internal resistance. L1 and L2 are the parameters of the induction coils. Find the quantity of heat discharged on the resistance R after the circuit is closed. Neglect all the other resistances in the circuit except those given.
Figure 5
Ideal pendulum (a small body on the end of a long weightless thread) with length l swings in the vertical plane with a small angular amplitude. In order to increase the amplitude of the swings the thread is being shortened by a small portion of its length Dl=l/100 as shown on the Fig. 6 in the vertical position of the pendulum. In the marginal positions of the pendulum the thread is being let loose by the same portion of its length. How many swings will take place before the amplitude of pendulum's swings will increase five times? Consider that the tension of the thread is constant at all times.
Figure 6
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