The gravitational force exerted by the planet Earth on a...

The gravitational force exerted by the planet Earth on a unit mass at a distance $r$ from the center of the planet is $$F(r)=\left\{\begin{array}{ll}{\frac{G M r}{R^{3}}} & {\text { if } r < R} \\ {\frac{G M}{I^{2}}} & {\text { if } r \geqslant R}\end{array}\right.$$ where $M$ is the mass of Earth, $R$ is its radius, and $G$ is the grav- itational constant. Is $F$ a continuous function of $r ?$

The gravitational force exerted by the planet Earth on a unit
mass at a distance $r$ from the center of the planet is
$$F(r)=\left\{\begin{array}{ll}{\frac{G M r}{R^{3}}} & {\text { if } r < R} \\ {\frac{G M}{I^{2}}} & {\text { if } r \geqslant R}\end{array}\right.$$
where $M$ is the mass of Earth, $R$ is its radius, and $G$ is the grav-
itational constant. Is $F$ a continuous function of $r ?$

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Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over various intervals of its domain. The study of these functions focuses on understanding properties such as continuity and differentiability, particularly at the junctures where the function’s rule changes.

Gravitational Field in Spherical Bodies

The gravitational field due to a spherical mass distribution is described by distinct expressions inside and outside the mass. This difference arises from the varying effects of mass distribution with distance from the center. The concepts behind these expressions are foundational in classical mechanics and astrophysics, illustrating how symmetry simplifies gravitational force calculations.

Continuity

Continuity examines whether a function has no breaks, jumps, or holes at any point in its domain. For piecewise functions, like the one in the example, checking continuity at the boundaries where the definition changes is essential. In such cases, the limits from both sides of the boundary must be the same as the function's value at that point to ensure overall continuity.

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