2018 USA Physics Olympiad Exam Part A 9 Question A3 A vacuum system consists of a chamber of volume V connected to a vacuum pump that is a cylinder with a piston that moves left and right.The minimum volume in the pump cylinder is Vo,and the maximum volume in the cylinder is Vo+AV.You should assume that Av<V. chamber motor cylinder outlet inlet piston The cylinder has two valves.The inlet valve opens when the pressure inside the cylinder is lower than the pressure in the chamber,but closes when the piston moves to the right.The outlet valve opens when the pressure inside the cylinder is greater than atmospheric pressure Pa,and closes when the piston moves to the left.A motor drives the piston to move back and forth.The piston moves at such a rate that heat is not conducted in or out of the gas contained in the cylinder during the pumping cycle.One complete cycle takes a time At.You should assume that At is a very small quantity,but AV/At=R is finite.The gas in the chamber is ideal monatomic and remains at a fixed temperature of Ta. Start with assumption that Vo =0 and there are no leaks in the system. a.At t=0 the pressure inside the chamber is Pa.Find an equation for the pressure at a later time t. Solution During each cycle,the system sucks gas out of the chamber and pushes it into the atmosphere. Since Vo =0,the inlet valve opens the moment the piston starts moving to the left.When the piston is all the way to the left,a fraction Av/V+AV)of the gas is in the cylinder.As the piston moves to the right,all of this gas is pushed out,so after a single cycle, (a) and in general, Pd)=卫 (a) While this is technically correct,it can be simplified significantly.Write P0=n(+）=(+ -t/△t where x=AV/V<1.Then using the definition of e, e=lim (1+2)1/ x→0 we have P(t)=Pae-Ri/V. Copyright C2018 American Association of Physics Teachers2018 USA Physics Olympiad Exam Part A 9 Question A3 A vacuum system consists of a chamber of volume V connected to a vacuum pump that is a cylinder with a piston that moves left and right. The minimum volume in the pump cylinder is V0, and the maximum volume in the cylinder is V0 + ∆V . You should assume that ∆V  V . V chamber cylinder outlet inlet piston motor The cylinder has two valves. The inlet valve opens when the pressure inside the cylinder is lower than the pressure in the chamber, but closes when the piston moves to the right. The outlet valve opens when the pressure inside the cylinder is greater than atmospheric pressure Pa, and closes when the piston moves to the left. A motor drives the piston to move back and forth. The piston moves at such a rate that heat is not conducted in or out of the gas contained in the cylinder during the pumping cycle. One complete cycle takes a time ∆t. You should assume that ∆t is a very small quantity, but ∆V /∆t = R is finite. The gas in the chamber is ideal monatomic and remains at a fixed temperature of Ta. Start with assumption that V0 = 0 and there are no leaks in the system. a. At t = 0 the pressure inside the chamber is Pa. Find an equation for the pressure at a later time t. Solution During each cycle, the system sucks gas out of the chamber and pushes it into the atmosphere. Since V0 = 0, the inlet valve opens the moment the piston starts moving to the left. When the piston is all the way to the left, a fraction ∆V /(V + ∆V ) of the gas is in the cylinder. As the piston moves to the right, all of this gas is pushed out, so after a single cycle, Pf = Pi  V V + ∆V  and in general, P(t) = Pa  V V + ∆V t/∆t . While this is technically correct, it can be simplified significantly. Write P(t) = Pa  1 + ∆V V −t/∆t = Pa  (1 + x) 1/x−Rt/V where x = ∆V /V  1. Then using the definition of e, e = lim x→0 (1 + x) 1/x we have P(t) = Pae −Rt/V . Copyright c 2018 American Association of Physics Teachers