Theoretical competition. Tuesday, 15 July 2014 1/1 Problem 1 (9 points) This problem consists of three independent parts. Part A (3 points) A small puck of mass is carefully placed onto the inner surface of the thin hollow thin cylinder of mass and of radius . Initially, the cylinder rests on the horizontal plane and the puck is located at the height above the plane as shown in the figure on the left. Find the interaction force between the puck and the cylinder at the moment when the puck passes the lowest point of its trajectory. Assume that the friction between the puck and the inner surface of the cylinder is absent, and the cylinder moves on the plane without slipping. The free fall acceleration is . Part B (3 points) A bubble of radius = 5.00 cm, containing a diatomic ideal gas, has the soap film of thickness = 10.0 m and is placed in vacuum. The soap film has the surface tension = 4.00 102 N m and the density = 1.10 g cm3. 1) Find formula for the molar heat capacity of the gas in the bubble for such a process when the gas is heated so slowly that the bubble remains in a mechanical equilibrium and evaluate it; 2) Find formula for the frequency of the small radial oscillations of the bubble and evaluate it under the assumption that the heat capacity of the soap film is much greater than the heat capacity of the gas in the bubble. Assume that the thermal equilibrium inside the bubble is reached much faster than the period of oscillations. Hint: Laplace showed that there is pressure difference between inside and outside of a curved surface, caused by surface tension of the interface between liquid and gas, so that = 2 . Part C (3 points) Initially, a switch is unshorted in the circuit shown in the figure on the right, a capacitor of capacitance 2 carries the electric charge 0, a capacitor of capacitance is uncharged, and there are no electric currents in both coils of inductance and 2, respectively. The capacitor starts to discharge and at the moment when the current in the coils reaches its maximum value, the switch is instantly shorted. Find the maximum current max through the switch thereafter. Theoretical competition. Tuesday, 15 July 2014 1/4 Problem 1 Solution Part A Consider the forces acting on the puck and the cylinder and depicted in the figure on the right. The puck is subject to the gravity force and the reaction force from the cylinder . The cylinder is subject to the gravity force , the reaction force from the plane 1, the friction force and the pressure force from the puck = . The idea is to write the horizontal projections of the equations of motion. It is written for the puck as follows = sin, (A.1) where is the horizontal projection of the puck acceleration. For the cylinder the equation of motion with the acceleration is found as = sin . (A.2) Since the cylinder moves along the plane without sliding its angular acceleration is obtained as = / (A.3) Then the equation of rotational motion around the center of mass of the cylinder takes the form = , (A.4) where the inertia moment of the hollow cylinder is given by = 2. (A.5) Solving (A.2)-(A.5) yields 2 = sin. (A.6) From equations (A.1) and (A.6) it is easily concluded that = 2. (A.7) Since the initial velocities of the puck and of the cylinder are both equal to zero, then, it follows from (A.7) after integrating that = 2. (A.8) It is obvious that the conservation law for the system is written as = 2 2 + 2 2 + 2 2 , (A.9) where the angular velocity of the cylinder is found to be = , (A.10) since it does not slide over the plane. Solving (A.8)-(A.10) results in velocities at the lowest point of the puck trajectory written as = 2 (2+), (A.12) = (2+). (A.13) In the reference frame sliding progressively along with the cylinder axis, the puck moves in a circle of radius and, at the lowest point of its trajectory, have the velocity = + (A.14) and the acceleration rel= rel 2 . (A.15) At the lowest point of the puck trajectory the acceleration of the cylinder axis is equal to zero, therefore, the puck acceleration in the laboratory reference frame is also given by (A.15). = 2 . (A.16) then the interaction force between the puck and the cylinder is finally found as = 3 1 + 3 . (A.17) Theoretical competition. Tuesday, 15 July 2014 2/4 Part B 1) According to the first law of thermodynamics, the amount of heat transmitted to the gas in the bubble is found as = + , (B.1) where the molar heat capacity at arbitrary process is as follows = 1 = + . (B.2) Here stands for the molar heat capacity of the gas at constant volu