Question

For motion near the surface of the earth, we usually assume that the gravitational force on a mass $m$ is $$\mathbf{F}=-mg\mathbf{k}$$ but for motion involving an appreciable variation in distance $r$ from the center of the earth, we must use $$\mathbf{F}=-\frac{C}{r^2}{\mathbf{e}}_r=-\frac{C}{r^2}\frac{\mathbf{r}}{|\mathbf{r}|}=-\frac{C}{r^3}\mathbf{r}$$ where $C$ is a constant. Show that both these $\mathbf{F}$ 's are conservative, and find the potential for each.

Short answer

Both forces are conservative because their curls are zero. The potential near Earth's surface is $V(r)=mgz+C$, and with varying distances, $V(r)=-\frac{C}{r}+C^{\prime }$.

Step-by-step solution

- Understand Gravitational Force Near Earth's Surface

The gravitational force on a mass near Earth's surface can be expressed as $\mathbf{F}=-mg\mathbf{k}.$The vector $\mathbf{k}$ represents the direction of the force (downwards, toward the center of Earth), and $g$ is the gravitational constant at the surface. This force is considered conservative if its curl is zero.

- Calculate Curl of $\mathbf{F}$ Near Earth's Surface

Calculate the curl of the force vector field: $abla\times (-mg\mathbf{k}).$Since $\mathbf{k}$ is a constant unit vector and $g$ is also constant near Earth's surface, $abla\times (-mg\mathbf{k})=\mathbf{0}.$This indicates that $\mathbf{F}=-mg\mathbf{k}$ is a conservative force.

- Find Potential for Gravitational Force Near Earth's Surface

A conservative force is derived from a potential function $V$: $\mathbf{F}=-ablaV.$Given $\mathbf{F}=-mg\mathbf{k}$, we find the potential function by integrating$-mg\mathbf{k}=-ablaV.$This implies, $V(r)=mgz+C.$Here, $V$ is the gravitational potential energy near Earth's surface, and $C$ is the integration constant.

- Understand Gravitational Force with Variation in Distance

The gravitational force for motion with appreciable variation in distance from Earth's center is given by $\mathbf{F}=-\frac{C}{r^2}\frac{\mathbf{r}}{|\mathbf{r}|}=-\frac{C}{r^3}\mathbf{r}$.

- Calculate Curl of $\mathbf{F}$ with Varying Distance

Since $\mathbf{r}$ represents the position vector, $\mathbf{F}=-\frac{C}{r^3}\mathbf{r}.$Evaluate the curl, $abla\times (-\frac{C}{r^3}\mathbf{r}).$In spherical coordinates, the curl of any central force derived from a radial potential is zero: $abla\times (-\frac{C}{r^3}\mathbf{r})=\mathbf{0}.$Thus, $\mathbf{F}$ is conservative.

- Find Potential for Force with Variation in Distance

Since $\mathbf{F}=-ablaV$, integrate $\mathbf{F}=-\frac{C}{r^3}\mathbf{r}$.The potential function $V$ is found by: $ablaV=\frac{C}{r^3}\mathbf{r}.$Integrating with respect to $r$ gives: $V(r)=-\frac{C}{r}+C^{\prime }.$Here, $C^{\prime }$ is the integration constant.

Key concepts

Gravitational Potential

The concept of gravitational potential is central to understanding how gravitational forces operate over different distances. Near the Earth's surface, the gravitational potential energy, denoted by $V$, is given by $V=mgz+C$, where $mgz$ represents the potential energy due to gravity at height $z$, and $C$ is an integration constant. Since gravitational forces are derived from potential energy, when you move an object in a gravitational field, you say it gains or loses gravitational potential energy. For instance, lifting a book higher increases its gravitational potential energy.
When you consider gravitational forces at distances where they vary significantly from the Earth's center, the gravitational potential is described differently. Here, the potential energy function is $V(r)=-\frac{C}{r}+C^{\prime }$, where $C^{\prime }$ is an integration constant and $r$ is the distance from the Earth's center. This formulation shows the inverse relationship between potential energy and distance.

Curl of a Vector Field

The curl of a vector field helps determine whether a force is conservative. A conservative force is significant because it means the work done moving an object between two points is independent of the path taken. This makes calculations much simpler. For a force to be conservative, the curl (or rotational part) of the force field, represented mathematically as $abla\times \mathbf{\text{F}}$, must be zero.
For example, near the Earth's surface, the gravitational force $\mathbf{\text{F}}=-mg\mathbf{\text{k}}$ is conservative because its curl is zero. This is why gravitational potential energy can be easily calculated as $V=mgz+C$. Similarly, in a varying distance scenario, where $\mathbf{\text{F}}=-\frac{C}{r^3}\mathbf{\text{r}}$, taking the curl again results in zero, solidifying the conservative nature of this force.

Potential Energy

Potential energy is the stored energy of an object due to its position in a force field, such as gravity. It is a fundamental concept because it converts into kinetic energy when the object moves. Near Earth's surface, potential energy (often called gravitational potential energy) can be given by $V=mgz+C$. This formulation is quite intuitive: the higher you lift an object, the more potential energy it has.
When considering varying distances from the Earth's center, the potential energy is expressed as $V(r)=-\frac{C}{r}+C^{\prime }$. It represents how gravitational energy changes with distance, showing forces diminish inversely with the square of the distance. Integrating these forces gives the potential energy, which helps to understand the energy dynamics of planets, satellites, and other celestial objects.
Understanding these potential energy expressions allows for predictions about the behavior of objects under the influence of varying gravitational forces. From dropping an apple to launching a satellite, these principles help explain numerous physical phenomena.