Problem 27 - Advanced Engineering Mathematics

Question

Evaluate \(\int_{C} \mathbf{F} \cdot d \mathbf{r}\). The inverse square law of gravitational attraction between two masses \(m_{1}\) and \(m_{2}\) is given by \(\mathbf{F}=-G m_{1} m_{2} \mathbf{r} /\|\mathbf{r}\|^{3}\), where \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Show that \(\mathbf{F}\) is conservative. Find a potential function for \(\mathbf{F}\).

Step-by-Step Solution

Answer

The vector field \(\mathbf{F}\) is conservative. The potential function is \(\phi = -\frac{Gm_1m_2}{\|\mathbf{r}\|}\).

Step 1: Verifying that \(\mathbf{F}\) is Conservative

To determine if \(\mathbf{F}\) is conservative, check if its curl is zero. The curl of a vector field \(\mathbf{F} = P\mathbf{i} + Q\mathbf{j} + R\mathbf{k}\) is given by:\[abla \times \mathbf{F} = \left(\frac{\partial R}{\partial y} - \frac{\partial Q}{\partial z}\right)\mathbf{i} + \left(\frac{\partial P}{\partial z} - \frac{\partial R}{\partial x}\right)\mathbf{j} + \left(\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\right)\mathbf{k}\]Substitute \(\mathbf{F} = -\frac{Gm_1m_2}{\|\mathbf{r}\|^3} \mathbf{r}\), and compute each partial derivative. It can be shown that each component of the curl is zero, confirming \(abla \times \mathbf{F} = \mathbf{0}\). Therefore, \(\mathbf{F}\) is conservative.

Step 2: Finding the Potential Function

Since \(\mathbf{F}\) is conservative, we can find a scalar potential function \(\phi\) such that \(\mathbf{F} = -abla \phi\). Given:\[\mathbf{F} = -\frac{G m_1 m_2}{\|\mathbf{r}\|^3} \mathbf{r},\]We look for \(\phi\) such that:\[\mathbf{F} = -\left( \frac{\partial \phi}{\partial x}\mathbf{i} + \frac{\partial \phi}{\partial y}\mathbf{j} + \frac{\partial \phi}{\partial z}\mathbf{k} \right).\]The gradient of the potential function is found by integrating each component of \(\mathbf{F}\) with respect to its respective coordinate. This gives:\[\phi = -\frac{G m_1 m_2}{\|\mathbf{r}\|}\]where \(\|\mathbf{r}\| = \sqrt{x^2 + y^2 + z^2}\).

Key Concepts

Inverse Square Law

The inverse square law is a fundamental principle seen in physics, especially in gravitational and electromagnetic fields. It states that the strength of an effect (such as force, light, or sound) radiating from a point source is inversely proportional to the square of the distance from that source.
In the context of gravitational force, this means that the gravitational attraction between two masses decreases as the square of the distance between them increases. This can be expressed mathematically by the formula for gravitational force:

• \(\mathbf{F} = -\frac{G m_1 m_2}{\|\mathbf{r}\|^3} \mathbf{r}\)

Here, \(G\) is the gravitational constant, \(m_1\) and \(m_2\) are the masses involved, and \(\mathbf{r}\) is the position vector representing the separation between the masses. This inverse relationship is crucial not only in calculating gravitational forces but also in ensuring accurate predictions of celestial mechanics.
As the distance between two objects doubles, for example, the gravitational pull between them becomes four times weaker. This powerful tool helps scientists and engineers design everything from satellite trajectories to understanding planetary motions.

Gravitational Force

Gravitational force is one of the most common manifestations of the inverse square law in nature. It's a fundamental force that attracts two bodies with mass. This force can be calculated using Newton's law of universal gravitation:

• \(\mathbf{F} = -\frac{G m_1 m_2}{\|\mathbf{r}\|^3} \mathbf{r}\)

Here, the force \(\mathbf{F}\) is directed along the vector \(\mathbf{r}\), which connects the center of the two masses \(m_1\) and \(m_2\). The negative sign indicates that the force is attractive, pulling the masses toward each other.
The role of the gravitational constant \(G\) is to scale the force appropriately, given the masses involved. This idea that two bodies can exert force on one another across a distance laid the foundation for understanding how planets and other celestial bodies interact in the cosmos.

Potential Function

A potential function is a scalar field whose spatial derivatives determine a vector field, particularly in a conservative field scenario. In simple terms, it's a function from which you can derive other functional properties, like force.
For gravitational force, the potential function \(\phi\) provides a simplified way to find the force. This function is derived by finding a scalar function \(\phi\) such that:

• \(\mathbf{F} = -abla \phi\)

For our gravitational field, we can derive the potential function using:

• \(\phi = -\frac{G m_1 m_2}{\|\mathbf{r}\|}\)

The potential function \(\phi\) encapsulates the work done by the force as an object moves through the field, offering a valuable tool for both qualitative and quantitative analysis in physics. Having this function allows for a deeper understanding of energy systems and how they behave. Moreover, the potential function simplifies calculations and reveals properties like energy conservation within a system, highlighting the elegance and interconnectedness of physical laws.