Sunday, September 21, 2008

Irodov Problem 1.216

Pressure is the derivative of component of the force acting in the normal direction to a surface with respect to the surface area (dF/dA). In this problem the gravitational force
due to the mass in the sphere causes the pressure at each layer of the sphere. The closer a certain layer to the center, the higher the pressure since the column of the mass pushing this layer is larger.

In the solution to problem 1.214 we already derived the gravitational field strength in a uniform sphere at a location that is r distance from the center as,

Here p is the density of the sphere. Since the gravitational field strength is the force per unit mass , the force acting on an infinitesimally small mass dm located at a distance r from the center is given by gdm. Consider an infinitesimally small cylindrical column of area dA located at a distance r from the center and height dr (as shown in the figure). The total mass of this infinitesimally small elemement is given by dm = pdAdr and so the gravitaional force acting on it is given by,

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