Work Done By A Force Field Along A Path Calculation And Explanation

In physics, the concept of work done by a force is fundamental. It quantifies the energy transferred when a force acts on an object causing displacement. When the force is constant and the displacement is along a straight line, the work done is simply the dot product of the force and displacement vectors. However, when dealing with variable forces or motion along a curved path, calculating work done requires a more sophisticated approach, often involving line integrals. This article delves into calculating the work done by a force field along a specific path, providing a step-by-step explanation and addressing common challenges.

This exploration is particularly relevant in fields like electromagnetism, fluid dynamics, and gravitational physics, where forces are often position-dependent and objects move along complex trajectories. Understanding how to compute work done in such scenarios is crucial for solving various problems, such as determining the energy required to move a charged particle in an electric field or analyzing the motion of a satellite in a gravitational field. The problem we will address involves finding the work done by the force field ${\vec{F} = (2xy + z)\hat{i} + x^2\hat{j} + x\hat{k}}$ along the path ${y = x^2}$ between the points (0,0,0) and (1,1,1). This problem exemplifies the application of line integrals in physics and provides a practical understanding of the underlying concepts.

To calculate the work done by a force field ${\vec{F}}$ along a path ${C}$, we use the line integral: ${W = \int_C \vec{F} \cdot d\vec{r},}$ where ${d\vec{r}}$ is an infinitesimal displacement vector along the path. This integral sums up the infinitesimal work done (${\vec{F} \cdot d\vec{r}}$) over the entire path. To evaluate this integral, we need to parameterize the path ${C}$ and express both ${\vec{F}}$ and ${d\vec{r}}$ in terms of a single parameter.

In our specific problem, the force field is given by ${\vec{F} = (2xy + z)\hat{i} + x^2\hat{j} + x\hat{k},}$ and the path is defined by ${y = x^2}$ between the points (0,0,0) and (1,1,1). This path lies in three-dimensional space, and we need to find a suitable parameterization to describe it. The parameterization will allow us to transform the line integral into a regular single-variable integral, which we can then evaluate using standard calculus techniques. The key is to choose a parameter that simplifies the calculations and accurately represents the path.

The conceptual framework involves several crucial steps:

1. Parameterizing the Path: Express the coordinates ${x}$, ${y}$, and ${z}$ as functions of a single parameter, say ${t}$. This parameterization should cover the path from the initial point (0,0,0) to the final point (1,1,1).
2. Expressing the Force Field in Terms of the Parameter: Substitute the parameterized coordinates into the expression for the force field ${\vec{F}}$. This will give ${\vec{F}}$ as a function of the parameter ${t}$.
3. Calculating the Differential Displacement Vector: Find ${d\vec{r}}$, which is given by ${d\vec{r} = \left(\frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}\right) dt.}$
4. Evaluating the Dot Product: Compute the dot product ${\vec{F} \cdot d\vec{r}}$, which will be a function of ${t}$.
5. Evaluating the Line Integral: Integrate the dot product with respect to ${t}$ over the appropriate limits, which correspond to the initial and final points of the path. This will give the total work done.

By following these steps systematically, we can accurately determine the work done by the force field along the specified path.

Let’s solve the problem step by step, starting with parameterizing the path.

1. Parameterizing the Path

The path is defined by ${y = x^2}$ between the points (0,0,0) and (1,1,1). Since ${y}$ is given as a function of ${x}$, it is natural to choose ${x}$ as the parameter. Let ${x = t}$. Then, since ${y = x^2}$, we have ${y = t^2}$. To find ${z}$, we note that the path connects (0,0,0) and (1,1,1), and since the path lies on the curve ${y = x^2}$, a simple parameterization for ${z}$ is ${z = x}$, so ${z = t}$ as well. This gives us the parameterization: ${\vec{r}(t) = t\hat{i} + t^2\hat{j} + t\hat{k},}$ where ${t}$ varies from 0 to 1 to cover the path from (0,0,0) to (1,1,1). This parameterization is crucial as it allows us to represent the path in terms of a single variable, simplifying the integral calculation.

2. Expressing the Force Field in Terms of the Parameter

The force field is given by ${\vec{F} = (2xy + z)\hat{i} + x^2\hat{j} + x\hat{k}.}$ Substituting ${x = t}$, ${y = t^2}$, and ${z = t}$ into the expression for ${\vec{F}}$, we get: ${\vec{F}(t) = (2t(t^2) + t)\hat{i} + t^2\hat{j} + t\hat{k} = (2t^3 + t)\hat{i} + t^2\hat{j} + t\hat{k}.}$ This step is essential because it transforms the force field, which was initially a function of ${x}$, ${y}$, and ${z}$, into a function of the single parameter ${t}$. This transformation is necessary for evaluating the line integral.

3. Calculating the Differential Displacement Vector

The differential displacement vector ${d\vec{r}}$ is given by: ${d\vec{r} = \left(\frac{dx}{dt}\hat{i} + \frac{dy}{dt}\hat{j} + \frac{dz}{dt}\hat{k}\right) dt.}$ Using our parameterization ${\vec{r}(t) = t\hat{i} + t^2\hat{j} + t\hat{k}}$, we find the derivatives: ${\frac{dx}{dt} = 1, \quad \frac{dy}{dt} = 2t, \quad \frac{dz}{dt} = 1.}$ Thus, ${d\vec{r} = (1\hat{i} + 2t\hat{j} + 1\hat{k}) dt.}$ Calculating ${d\vec{r}}$ is a key step in setting up the line integral. It represents an infinitesimal displacement along the parameterized path, allowing us to integrate the force along the curve.

4. Evaluating the Dot Product

Now, we compute the dot product ${\vec{F}(t) \cdot d\vec{r}}$: ${\vec{F}(t) \cdot d\vec{r} = ((2t^3 + t)\hat{i} + t^2\hat{j} + t\hat{k}) \cdot (1\hat{i} + 2t\hat{j} + 1\hat{k}) dt.}$ Taking the dot product, we get: ${\vec{F}(t) \cdot d\vec{r} = (2t^3 + t)(1) + (t^2)(2t) + (t)(1) dt = (2t^3 + t + 2t^3 + t) dt = (4t^3 + 2t) dt.}$ The dot product ${\vec{F}(t) \cdot d\vec{r}}$ represents the infinitesimal work done by the force field along the infinitesimal displacement ${d\vec{r}}$. This step simplifies the problem to a single-variable integral.

5. Evaluating the Line Integral

Finally, we evaluate the line integral to find the total work done: ${W = \int_C \vec{F} \cdot d\vec{r} = \int_{0}^{1} (4t^3 + 2t) dt.}$ Integrating with respect to ${t}$, we have: ${W = \left[t^4 + t^2\right]_{0}^{1} = (1^4 + 1^2) - (0^4 + 0^2) = 1 + 1 = 2.}$ Therefore, the work done by the force field along the path ${y = x^2}$ between (0,0,0) and (1,1,1) is 2 units. This result is obtained by systematically parameterizing the path, expressing the force field in terms of the parameter, calculating the differential displacement vector, evaluating the dot product, and finally, evaluating the line integral.

In conclusion, we have successfully calculated the work done by the force field ${\vec{F} = (2xy + z)\hat{i} + x^2\hat{j} + x\hat{k}}$ along the path ${y = x^2}$ between the points (0,0,0) and (1,1,1). The work done is 2 units. This problem demonstrates the importance of line integrals in calculating the work done by a force along a curved path. The key steps involve parameterizing the path, expressing the force field in terms of the parameter, computing the differential displacement vector, evaluating the dot product, and integrating over the parameter range.

The systematic approach used in this solution can be applied to various problems in physics and engineering where the calculation of work done by a force field along a specified path is required. Understanding these concepts and techniques is crucial for advanced studies in fields like electromagnetism, fluid dynamics, and classical mechanics.

By mastering the techniques discussed in this article, students and professionals can confidently tackle problems involving variable forces and curved paths, enhancing their problem-solving skills in physics and related disciplines. The ability to calculate work done accurately is essential for understanding energy transfer and system dynamics in a wide range of physical phenomena.