A bicycle tire that has a slow leak,

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    Bicycle Tire
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SUMMARY

The discussion revolves around estimating the size of a hole in a bicycle tire that causes it to go flat within an hour. The relevant equations include N(t)=N(0)exp(-t/tau) and dN/dt=-A/2V*sqrt(kT/m)*N, where tau represents the characteristic time. Participants express confusion regarding the integration of these equations and the implications of the tire's volume decreasing to zero. Clarification is sought on the derivation of the second equation and its applicability to the problem at hand.

PREREQUISITES
  • Understanding of exponential decay functions and their applications.
  • Familiarity with gas laws and the kinetic theory of gases.
  • Basic knowledge of calculus, particularly integration techniques.
  • Concept of characteristic time in physical systems.
NEXT STEPS
  • Research the derivation of the equation dN/dt=-A/2V*sqrt(kT/m)*N in the context of gas leakage.
  • Study the implications of volume changes in dynamic systems, particularly in gas behavior.
  • Explore practical methods for measuring tire pressure and leak rates.
  • Learn about the application of exponential decay in real-world scenarios, such as tire maintenance.
USEFUL FOR

Students studying physics or engineering, particularly those focusing on thermodynamics and fluid dynamics, as well as anyone interested in practical applications of gas laws in everyday scenarios like bicycle maintenance.

pentazoid
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Homework Statement


If you poke a hole in a container full of gas, the gas will start leaking out. In this problem you will make a rough estimate of the rate at which gas escapes through a hole.

Your bicycle tire has a slow leak, so that it goes flat within about an hour after being inflate. Rougly how big is the hole? (use any reasonable estimate for the volume of the tire.)


Homework Equations



N(t)=N(0)exp(-t/tau), tau being the characteristic time.
dN/dt=-A/2V*sqrt(kT/m)*N

The Attempt at a Solution



By rouglhly how big is the hole, do they want me to guesstimate the volume of the tire?
 
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Did anyone not understand my question or solution?
 
I did not. It's not clear to me what is meant by the tire becoming flat. V=0?
In this case, volume should be a variable. Which should mean that you can't integrate the second equation and get the first one. So they contradict each other. Can you tell me how the second equation is derived? That is, what is the situation for which it holds?
 

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