ES 123 Problem Set 6 Solution Lihua Jin Mar 26, 2010 1. According to the mass conservation, we have 00VVρρ=(1) where ρand Vare the density and volume at any time. Since the cross-section area keeps a constant, we can cancel area from Eq. (1), i.e. 00LLρρ=(2) where we know 0LLvt=−(3) so the dependence of the density on time is 000LLvtρρ=−(4) You can also use the differential equation of mass conservation to solve this problem. Actually, Eq. (1) is the ‘integral’ form, which is similar to the integral momentum conservation law we talked about in class. Eq. (1) can be derived from the differential equation of mass conservation. Concept question: Assume the air in the cylinder is an ideal gas, which satisfies pRTMρ=(5) so the force is proportional to the pressure and can be calculated as ()000L RAMTFpALvtρ==−(6) which can be sketched as shown in Fig. 1. The force increases with the time. Fig. 1
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