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

Answer:

Yes, F is a continuous function of r

Step-by-step explanation:

We are given that

When r<R

$F(r)=\frac{GMr}{R^3}$

$r\geq R$

$F(r)=\frac{GM}{r^2}$

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

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

RHL

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

$F(R)=\frac{GM}{R^2}$

When a function is continuous at x=a

Then, LHL=RHL=f(a)

RHL==LHL=F(R)

Hence, the function is continuous of r.