2014 USA Physics Olympiad Exam Part A 3 Part A Question Al Inspired by:http://www.wired.com/wiredscience/2012/04/a-leaning-motorcycle-on-a-vertical-wall/ A unicyclist of total height h goes around a circular track of radius R while leaning inward at an angle 0 to the vertical.The acceleration due to gravity is g. a.Suppose h<R.What angular velocity w must the unicyclist sustain? b.Now model the unicyclist as a uniform rod of length h,where h is less than R but not negligible.This refined model introduces a correction to the previous result.What is the new expression for the angular velocity w?Assume that the rod remains in the plane formed by the vertical and radial directions,and that R is measured from the center of the circle to the point of contact at the ground. 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. pump heating coil 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. b.The room is insulated,but heat still passes into the room at a rate R=kAT,where AT is the temperature difference between the inside and the outside of the room and k is a constant. Find the coldest possible temperature of the room in terms of TH,k,and P. c.A typical room has a value of k =173 W/C.If the outside temperature is 40C,what minimum power should the air conditioner have to get the inside temperature down to 25C? Copyright C2014 American Association of Physics Teachers2014 USA Physics Olympiad Exam Part A 3 Part A Question A1 Inspired by: http://www.wired.com/wiredscience/2012/04/a-leaning-motorcycle-on-a-vertical-wall/ A unicyclist of total height h goes around a circular track of radius R while leaning inward at an angle θ to the vertical. The acceleration due to gravity is g. a. Suppose h  R. What angular velocity ω must the unicyclist sustain? b. Now model the unicyclist as a uniform rod of length h, where h is less than R but not negligible. This refined model introduces a correction to the previous result. What is the new expression for the angular velocity ω? Assume that the rod remains in the plane formed by the vertical and radial directions, and that R is measured from the center of the circle to the point of contact at the ground. 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. b. The room is insulated, but heat still passes into the room at a rate R = k∆T, where ∆T is the temperature difference between the inside and the outside of the room and k is a constant. Find the coldest possible temperature of the room in terms of TH, k, and P. c. A typical room has a value of k = 173 W/ ◦C. If the outside temperature is 40◦C, what minimum power should the air conditioner have to get the inside temperature down to 25◦C? Copyright c 2014 American Association of Physics Teachers