Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
Consider a thin parcel of air at an arbitrary height z in the atmosphere (z= 0 corresponds to sea-level). It has thickness Az, cross-sectional area A, and mass density p. Assume gravity (g) is constant, and there is no wind, so the parcel is neither rising nor falling. There are three relevant forces: - The upward force F, acting on the bottom of the parcel. This is caused by air pressure P, from below. - The downward force F, acting on the top of the parcel. This is caused by air pressure P; from above. The weight force of the parcel itself, due to gravity.

Consider a thin parcel of air at an arbitrary height z in the atmosphere (z= 0 corresponds to sea-level). It has thickness Az, cross-sectional area A, and mass density p. Assume gravity (g) is constant, and there is no wind, so the parcel is neither rising nor falling. There are three relevant forces: - The upward force F, acting on the bottom of the parcel. This is caused by air pressure P, from below. - The downward force F, acting on the top of the parcel. This is caused by air pressure P; from above. The weight force of the parcel itself, due to gravity.

Question

I need to know how to solve this ODE IVP set 

Consider a thin parcel of air at an arbitrary height z in the atmosphere (z= 0 corresponds to sea-level). It has thickness Az, cross-sectional area A, and mass density p. Assume gravity (g) is constant, and there is no wind, so the parcel is neither rising nor falling. There are three relevant forces: The upward force F, acting on the bottom of the parcel. This is caused by air from below. pressure Pp - The downward force F, acting on the top of the parcel. This is caused by air pressure P; from above. The weight force of the parcel itself, due to gravity. Fi Az F9
Transcribed Image Text:Consider a thin parcel of air at an arbitrary height z in the atmosphere (z= 0 corresponds to sea-level). It has thickness Az, cross-sectional area A, and mass density p. Assume gravity (g) is constant, and there is no wind, so the parcel is neither rising nor falling. There are three relevant forces: The upward force F, acting on the bottom of the parcel. This is caused by air from below. pressure Pp - The downward force F, acting on the top of the parcel. This is caused by air pressure P; from above. The weight force of the parcel itself, due to gravity. Fi Az F9
1. If AP = Pt – P, show that AP = -pgAz (Hint: Recall that force and pressure are related by the equation F=P A, where A is the area on which the pressure is acting.) 2. By taking the limit as the parcel's thickness goes to zero, establish a separable ODE relating pressure to height. (Note that density may in general depend on pressure) 3. According to the ideal gas law, pressure P, temperature T, and density p are all related as P = BpT, where B is a constant. Using this information, modify the ODE you found in part b). Note that temperature may depend on height. 4. Solve the IVP, where the pressure at sea level is Po. In this part of the question, assume temperature is constant (T= To). 5. Solve the IVP assuming temperature is To at the surface of the Earth, but drops linearly with altitude.
Transcribed Image Text:1. If AP = Pt – P, show that AP = -pgAz (Hint: Recall that force and pressure are related by the equation F=P A, where A is the area on which the pressure is acting.) 2. By taking the limit as the parcel's thickness goes to zero, establish a separable ODE relating pressure to height. (Note that density may in general depend on pressure) 3. According to the ideal gas law, pressure P, temperature T, and density p are all related as P = BpT, where B is a constant. Using this information, modify the ODE you found in part b). Note that temperature may depend on height. 4. Solve the IVP, where the pressure at sea level is Po. In this part of the question, assume temperature is constant (T= To). 5. Solve the IVP assuming temperature is To at the surface of the Earth, but drops linearly with altitude.
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer