The gravitational force exerted by the earth on a unit mass...

The gravitational force exerted by the earth on a unit mass
at a distance $r$ from the center of the planet is

$$F(r)=\left\{\begin{array}{ll}{\frac{G M r}{R^{3}}} & {\text { if } r < R} \\ {\frac{G M}{r^{2}}} & {\text { if } r \geqslant R}\end{array}\right.$$

Key Concepts

Newton's Law of Gravitation

This law states that every two point masses attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. It forms the foundation for understanding gravitational interactions and how mass influences space.

Gravitational Field Inside a Spherical Body

Within a uniformly dense sphere, like an idealized planet, the gravitational force on a body is directly proportional to its distance from the center. This results from the fact that only the mass enclosed within that radius contributes to the gravitational pull, leading to a linear increase in force with distance inside the sphere.

Gravitational Field Outside a Spherical Body

Outside a spherically symmetric mass distribution, the gravitational force obeys the inverse-square law. This means that the force diminishes proportionally to the square of the distance from the center, as the mass can be treated as if it were concentrated at a single point, greatly simplifying gravitational calculations.

Shell Theorem

The shell theorem is a crucial concept which states that a uniform spherical shell of mass produces no net gravitational force on an object located inside it, while for objects outside, its gravitational influence is equivalent to having all the mass concentrated at its center. This theorem simplifies the analysis of gravitational fields in spherically symmetric bodies.

Uniform Density Distribution

Assuming a uniform density for a spherical body simplifies the computation of gravitational forces. It allows the interior gravitational field to be expressed as a function of the distance from the center by considering only the mass enclosed within that radius, which is key when examining gravitational forces inside the sphere.

Transcript

00:01 Well, let's now see this function, f of r, sorry, f of r equal.

00:15 So it's g capital m r over capital r cubed, if r is r is smaller than capital r or it's g m over r squared if r is greater or equal capital r.

00:53 Well, this is the gravitational force exerted exerted by the earth on a unit mass at a distance r from the center of the planet.

01:17 And of course capital r is just the rth radius radius.

01:37 So let's prove that this function is continuous.

01:46 So what should we prove? well, we should prove that f is continuous, of course, at small r equal capital r.

02:02 So it's continuous when the function is evaluated at capital r, when the radius we are taking into account, when the distance we are taking into current is equal to the radius of the earth.

02:26 So, first of all, so we want to prove this that is continuous at capital r.

02:38 So first of of all, the value of the function at capital r is equal to when it's just g, capital m, g is the gravitational constant, capital m is the half mass over, sorry, over the radius squared.

03:05 That's f of capital now let's see the limit.

03:12 Limit one smaller approaches capital r.

03:19 So what does that mean? when we approach the radius of the earth, either from inside the earth or from outside, which means we are just on the surface of the earth, but just outside of it.

03:41 So let's see if this limit exists...

Please give Ace some feedback