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 $$F(r)=\left\{\begin{array}{ll}{\frac{G M r}{R^{3}}} & {\text { if } r < R} \\ {\frac{G M}{r^{2}}} & {\text { if } r \geqslant R}\end{array}\right.$$ where $M$ is the mass of Earth, $R$ is its radius, and $G$ is the gravitational 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
$$F(r)=\left\{\begin{array}{ll}{\frac{G M r}{R^{3}}} & {\text { if } r < R} \\ {\frac{G M}{r^{2}}} & {\text { if } r \geqslant R}\end{array}\right.$$
where $M$ is the mass of Earth, $R$ is its radius, and $G$ is the
gravitational constant. Is $F$ a continuous function of $r ?$

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Key Concepts

Gravitational Field Inside and Outside a Spherical Body

For a spherically symmetric mass distribution, the gravitational force behaves differently inside and outside the sphere. Inside a uniform sphere, the force increases linearly with distance from the center, whereas outside, it follows the inverse square law. This distinction is derived from using symmetry and principles such as Gauss's law adapted for gravity.

Newton's Law of Gravitation

This law states that any two masses attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between their centers. It forms the basis for calculating gravitational forces in both uniform and non-uniform fields.

Continuity of Functions

A function is considered continuous at a point if there is no abrupt jump at that point; specifically, the limit of the function as it approaches the point from the left equals the function's value at that point, which in turn equals the limit from the right. When a function is defined piecewise, ensuring it is continuous across its domains involves verifying these conditions at the boundaries between pieces.

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