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 / R^3 if r < R
GM / r^2 if r ≥ R
where M is the mass of Earth, R is its radius, and G is the gravitational constant.
1. Is F a continuous function of r?

Answer 1

Yes, F is a continuous function of r

Explanation:

We are given that

When r<R

`F(r)=(GMr)/(R^3)`

`r\geq R`

`F(r)=(GM)/(r^2)`

Where M=Mass of the earth

R=Radius of earth

G=Gravitational constant

We have to find the function is continuous of r or not.

LHL

`\lim_(r\rightarrow R-)(GMr)/(R^3)=(GMR)/(R^3)=(GM)/(R^2)`

RHL

`\lim_(r\rightarrow R+)(GM)/(R^2)=(GM)/(R^2)`

`F(R)=(GM)/(R^2)`

When a function is continuous at x=a

Then, LHL=RHL=f(a)

RHL==LHL=F(R)

Hence, the function is continuous of r.