Problem 30 - Advanced Engineering Mathematics

Question

A particle in the plane is attracted to the origin with a force \(\mathbf{F}=\|\mathbf{r}\|^{n} \mathbf{r}\), where \(n\) is a positive integer and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}\) is the position vector of the particle. Show that \(\mathbf{F}\) is conservative. Find the work done in moving the particle between \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right) .\)

Step-by-Step Solution

Answer

The force is conservative. Work done is given by change in potential energy: \(W = V(x_1, y_1) - V(x_2, y_2)\).

Step 1: Understanding Conservation

To determine if a force \(\mathbf{F}\) is conservative, it must be the gradient of some scalar potential function \(V\), i.e., \(\mathbf{F} = -abla V\). Additionally, the curl of \(\mathbf{F}\) must be zero: \(abla \times \mathbf{F} = \mathbf{0}\).

Step 2: Define the Force

The given force is \(\mathbf{F} = \|\mathbf{r}\|^n \mathbf{r}\), where \(\mathbf{r} = x \mathbf{i} + y \mathbf{j}\). The magnitude \(\|\mathbf{r}\| = \sqrt{x^2 + y^2}\), so \(\mathbf{F} = (x^2 + y^2)^{n/2} (x \mathbf{i} + y \mathbf{j})\).

Step 3: Compute the Curl of \(\mathbf{F}\)

To show \(\mathbf{F}\) is conservative, calculate the curl: \(abla \times \mathbf{F} = \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \mathbf{k}\). If \(abla \times \mathbf{F} = \mathbf{0}\), \(\mathbf{F}\) is conservative.

Step 4: Verify Zero Curl Condition

Compute \(F_x = \|\mathbf{r}\|^n x\) and \(F_y = \|\mathbf{r}\|^n y\). Then calculate the partial derivatives: \(\frac{\partial F_y}{\partial x}\) and \(\frac{\partial F_x}{\partial y}\). You will find that they are equal, confirming \(abla \times \mathbf{F} = \mathbf{0}\). Thus, \(\mathbf{F}\) is conservative.

Step 5: Define a Potential Function

Since \(\mathbf{F}\) is conservative, there exists a potential function \(V(x, y)\). We find \(V\) such that \(-abla V = \mathbf{F}\). Solve the partial differential equations \(\frac{\partial V}{\partial x} = -F_x\) and \(\frac{\partial V}{\partial y} = -F_y\).

Step 6: Calculate Work Done

The work done by a conservative force is the difference in potential energy: \(W = V(x_1, y_1) - V(x_2, y_2)\). Use the potential function \(V\) found in the previous step to calculate work done as the particle moves from \((x_1, y_1)\) to \((x_2, y_2)\).

Key Concepts

Gradient

The gradient is a crucial concept in vector calculus, and it helps us understand how a function changes in space. When you think of the gradient, imagine an arrow pointing towards the direction of the greatest rate of increase of a function.

• For a function of two variables, like a scalar potential function, the gradient is a vector that represents how those function values change across the x and y axes.
• The mathematical representation of the gradient operator is \( abla \). In two dimensions, \( abla = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y} \right) \).
• When a force is described as conservative, it means that this force can be expressed as the negative gradient of a potential function, \( \mathbf{F} = -abla V \).

This formulation plays a significant role in determining the conservation of energy in a system and finding potential energy functions.

Potential Function

A potential function, denoted as \(V(x, y)\), is an essential concept in the study of conservative forces. This function helps us describe the energy landscape of a force field.

• Conservative forces have potential functions, where the force exerted by a field is the negative gradient of the potential function, i.e., \( \mathbf{F} = - abla V \).
• In simpler terms, this means that a potential function gives us a way to describe the energy required to move within a field.
• To find the potential function, we solve differential equations based on the components of the force: \( \frac{\partial V}{\partial x} = - F_x \) and \( \frac{\partial V}{\partial y} = - F_y \).

Finding a potential function helps us understand energy conservation and predict the behavior of particles in a field. It enables us to compute things like the work done by these forces as particles move from one point to another.

Curl

The curl of a vector field helps determine if a field is conservative. In two dimensions, calculating the curl involves partial derivatives of the components of the vector field.

• The curl measures the rotation or 'circulation' within a field. For a vector field \( \mathbf{F} = (F_x, F_y) \), the curl is \( abla \times \mathbf{F} = \left( \frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y} \right) \mathbf{k} \).
• If the curl is zero, \( abla \times \mathbf{F} = \mathbf{0} \), the field is irrotational and likely conservative.
• Checking the curl is part of verifying the conservative nature of a force field because if a field can be derived from a potential function, its curl must be zero.

This property is essential in physics and engineering for simplifying complex problems and ensuring energy conservation.

Work Done

The concept of work done by a force is central to energy transfer and conservation. When a force moves an object, energy is transferred, which we quantify as work.

• For conservative forces, the work done is path-independent and only depends on the initial and final positions, making calculations straightforward.
• The work done by a conservative force is the difference between the potential energy at the starting point and the potential energy at the endpoint: \( W = V(x_1, y_1) - V(x_2, y_2) \).
• In other words, the work is simply the change in potential energy as the particle moves through the field.

Understanding work done in the context of conservative forces and potential energy is crucial for problems involving mechanical energy, allowing one to solve for unknown quantities and predict system behavior accurately.