Problem 27 - Advanced Engineering Mathematics

Question

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

\( \mathbf{F} \) is conservative; potential function is \( V = -G \frac{m_{1} m_{2}}{\sqrt{x^2 + y^2 + z^2}} \).

Step 1: Understand the Inverse Square Law

The formula for gravitational attraction, \( \mathbf{F} = -G \frac{m_{1} m_{2} \mathbf{r}}{\|\mathbf{r}\|^3} \), states that the force \( \mathbf{F} \) is a vector given by the product of the gravitational constant \( G \), the masses \( m_1 \) and \( m_2 \), and the direction vector \( \mathbf{r} \). This vector is divided by the cube of the magnitude of \( \mathbf{r} \). \( \mathbf{r} \) can be expressed in terms of its components: \( \mathbf{r} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} \).

Step 2: Determine if \( \mathbf{F} \) is Conservative

A vector field is conservative if it can be written as the negative gradient of a scalar potential function. Mathematically, for \( \mathbf{F} \) to be conservative, there should exist a function \( V(x, y, z) \) such that \( \mathbf{F} = -abla V \). This implies that the curl of \( \mathbf{F} \) should be zero, i.e., \( abla \times \mathbf{F} = \mathbf{0} \).

Step 3: Verify the Curl of \( \mathbf{F} \) is Zero

To verify that \( abla \times \mathbf{F} = \mathbf{0} \), calculate the curl \( abla \times \left( -G \frac{m_{1} m_{2} \mathbf{r}}{\|\mathbf{r}\|^3} \right) \). Using known properties about central forces and inverse square laws, it is established that the curl of such a force field is zero, confirming that \( \mathbf{F} \) is conservative.

Step 4: Find the Potential Function \( V \)

Since \( \mathbf{F} = -G \frac{m_{1} m_{2} \mathbf{r}}{\|\mathbf{r}\|^3} \) is conservative, find the potential function \( V \) such that \( abla V = -\mathbf{F} \). Integrating the radial component we get: \( V(\mathbf{r}) = -G \frac{m_{1} m_{2}}{\|\mathbf{r}\|} = -G \frac{m_{1} m_{2}}{\sqrt{x^2 + y^2 + z^2}} \). This function \( V \) represents the potential energy due to gravitational attraction.

Key Concepts

Inverse Square Law

The inverse square law is a fundamental principle in physics that describes how certain forces behave. Specifically, it states that the strength of a force diminishes with the square of the distance from the source. For gravitational forces between two masses, this is mathematically represented as:\[\mathbf{F} = -G \frac{m_{1} m_{2} \mathbf{r}}{\|\mathbf{r}\|^3}\]Here, \( G \) is the gravitational constant, \( m_1 \) and \( m_2 \) are the masses involved, and \( \mathbf{r} \) is the position vector pointing towards one mass. Since it is divided by \( \|\mathbf{r}\|^3 \), this shows that the force decreases rapidly as the distance increases. This law applies to other forces, like those in electrostatics, highlighting its broad importance. By understanding this law, you can see how gravitational force is weaker the further you are from a mass, changing noticeably as distance grows.

Gravitational Attraction

Gravitational attraction is the force that pulls two masses toward each other. It's a key concept describing the interaction between masses everywhere in the universe. Derived from Newton's law of universal gravitation, the force of attraction between two objects depends on the mass of both objects and the distance between them. The equation, as shown previously, is:\[\mathbf{F} = -G \frac{m_{1} m_{2} \mathbf{r}}{\|\mathbf{r}\|^3}\]Important points about gravitational attraction:

• It's always attractive, never repulsive.
• The force is proportional to the product of the two masses: As mass increases, so does the force.
• The force decreases with the inverse square of the distance between the centers of the two masses.
• This attraction is what keeps planets orbiting stars and moons orbiting planets.

Recognizing how gravitational forces work helps explain many physical phenomena, from the tides to the orbits of planets.

Potential Function

A potential function is essential in understanding conservative forces, like the gravitational force. It is a scalar function whose gradient yields the force itself. Essentially, it tells us how energy is distributed in a field. For a conservative force field \( \mathbf{F} \), there is a potential function \( V \) such that:\[\mathbf{F} = -abla V\]This means the force \( \mathbf{F} \) is the negative gradient of \( V \). For gravitational attraction, the potential function is:\[V(\mathbf{r}) = -G \frac{m_{1} m_{2}}{\|\mathbf{r}\|} = -G \frac{m_{1} m_{2}}{\sqrt{x^2 + y^2 + z^2}}\]Key points about potential functions:

• It helps to calculate potential energy, which is crucial for understanding energy conservation in a system.
• Potential energy indicates the work done by or against the gravitational force.
• Understanding the potential function allows us to see how changes in position affect potential energy, useful in many mechanical problems.

By knowing the potential function, predicting motion and understanding energy dynamics in gravitational fields becomes much easier.