Question

) The acceleration due to gravity, g, varies with height above the surface of the earth, in a certain way. If you go down below the surface of the earth, g varies in a different way. It can be shown that g is given by Here R is the radius of the earth, M is the mass of the earth, G is the gravitational constant, and r is the distance to the center of the earth.

Sketch a graph of g against r, and use it to answer the following questions:

A. Is g a continuous function of r? why?


B. Is g a differentiable function of r? why?

Answer 1

Please close the question. Post the Question in the correct Section.

Answer 2

Anyways,
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Answer 3

When \(R<R_e\), \(g=\cfrac{GM}{R^2}=\cfrac{4\pi R^3\rho}{R^2}=\cfrac{4}{3}G\pi R \rho\), where \(\rho\) is the density of the earth, \(G\) is the gravitational constant, \(R_e\) is the radius of the earth and \(M\) is the mass of the earth. In this case, \(g\) varies \(linearly\) with \(R\).


When \(R>R_e\), \(g=\cfrac{GM}{R^2}\) and \(g\) varies in an inverse square fashion with \(R\):


You can check if both are continuous and differentiable at \(R=R_e\), which would be the only points of concern. Use the definition of \(limit\).