Question

The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is F(r) = GMr R3 if r < R GM r2 if r ≥ R where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r? Why/why not?

Answer 1

Answer:

Yes the function is continuous.

Step-by-step explanation:

Given

$F(r)=\frac{GMr}{R^{3}}(r< R)$ and $F(r)=\frac{GM}{r^{2}}(r\geq R)$

For a function to be continuous we should have

$\lim_{r\rightarrow R^{-}}F(r)=\lim_{r\rightarrow R^{+}}F(r)$

Thus

$\lim_{r\rightarrow R^{-}}F(r)=\frac{GMR}{R^{3}}=\frac{GM}{R^{2}}$

Similarly we have

$\lim_{r\rightarrow R^{+}}F(r)=\frac{GM}{R^{2}}$

thus we can see that

$\lim_{r\rightarrow R^{-}}F(r)=\lim_{r\rightarrow R^{+}}F(r)$

Hence the function is continuous.