2014 USA Physics Olympiad Exam Part A 4 h Te mw2h cos 0 R 2-3sin Gravity continues to act at the center of mass,a distancefrom the point of contact,and in the opposite direction: Ta=-mg2 sin0 Again,the total torque is zero,so mw?h cos 0 R h h 、23sin0 mg sin0=0 tan 2h 3R sin0 Question A2 A room air conditioner is modeled as a heat engine run in reverse:an amount of heat QL is absorbed from the room at a temperature TL into cooling coils containing a working gas;this gas is compressed adiabatically to a temperature TH;the gas is compressed isothermally in a coil outside the house,giving off an amount of heat QH;the gas expands adiabatically back to a temperature TL;and the cycle repeats.An amount of energy W is input into the system every cycle through an electric pump.This model describes the air conditioner with the best possible efficiency. heating coil pump room valve cooling coil Assume that the outside air temperature is TH and the inside air temperature is TL.The air-conditioner unit consumes electric power P.Assume that the air is sufficiently dry so that no condensation of water occurs in the cooling coils of the air conditioner.Water boils at 373 K and freezes at 273 K at normal atmospheric pressure. a.Derive an expression for the maximum rate at which heat is removed from the room in terms of the air temperatures TH,TL,and the power consumed by the air conditioner P.Your derivation must refer to the entropy changes that occur in a Carnot cycle in order to receive full marks for this part. Copyright C2014 American Association of Physics Teachers2014 USA Physics Olympiad Exam Part A 4 τc = mω2h cos θ  R 2 − h 3 sin θ  Gravity continues to act at the center of mass, a distance h 2 from the point of contact, and in the opposite direction: τg = −mg h 2 sin θ Again, the total torque is zero, so mω2h cos θ  R 2 − h 3 sin θ  − mg h 2 sin θ = 0 ω = s  g R tan θ   1 − 2 3 h R sin θ −1 Question A2 A room air conditioner is modeled as a heat engine run in reverse: an amount of heat QL is absorbed from the room at a temperature TL into cooling coils containing a working gas; this gas is compressed adiabatically to a temperature TH; the gas is compressed isothermally in a coil outside the house, giving off an amount of heat QH; the gas expands adiabatically back to a temperature TL; and the cycle repeats. An amount of energy W is input into the system every cycle through an electric pump. This model describes the air conditioner with the best possible efficiency. room cooling coil pump valve heating coil Assume that the outside air temperature is TH and the inside air temperature is TL. The air-conditioner unit consumes electric power P. Assume that the air is sufficiently dry so that no condensation of water occurs in the cooling coils of the air conditioner. Water boils at 373 K and freezes at 273 K at normal atmospheric pressure. a. Derive an expression for the maximum rate at which heat is removed from the room in terms of the air temperatures TH, TL, and the power consumed by the air conditioner P. Your derivation must refer to the entropy changes that occur in a Carnot cycle in order to receive full marks for this part. Copyright c 2014 American Association of Physics Teachers