Question

A particle moves along the \(x\) axis under the influence of a conservative force that is described by $$\overrightarrow{\mathbf{F}}=-\alpha x e^{-\beta x^{2}} \hat{\mathbf{i}}$$ where \(\alpha\) and \(\beta\) are constants. Find the potential energy function \(U(x)\).

Short answer

The potential energy function will be \(U(x) = -\alpha e^{-\beta x^{2}}\).

Step-by-step solution

Observe the Force Component

Notice that the force \(\overrightarrow{\mathbf{F}}\) is only along the x-axis and its expression is given by \(-\alpha x e^{-\beta x^{2}}\). The \(\hat{\mathbf{i}}\) indicates the direction of the force, which in this case is in the x-direction.

Implement the Relation Between Force and Potential Energy

Recall that the force is the negative gradient of potential energy. Hence, in terms of potential energy \(U(x)\), the force can be expressed as \(\overrightarrow{\mathbf{F}} = - \nabla{U(x)}\), where \(\nabla\) is the nabla operator symbolizing a vector differential. Since the force is acting only in the x-direction, this can be simplified to \(\overrightarrow{\mathbf{F}} = -\frac{d}{dx} U(x)\).

Solve for the Potential Energy Function

Now integrate the force equation with respect to \(x\) to solve for \(U(x)\). This will yield: \(U(x) = - \int -\alpha x e^{-\beta x^{2}} dx\). To solve this type of integral, make a substitution, let \(u = \beta x^{2}\). This will convert the integral into the form: \(U(x) = \alpha \int e^{-u} du\). This integral results in \(-\alpha e^{-u}\). Now, revert back to the x variable by substituting \(u\) with \(\beta x^{2}\). Hence, the potential energy function will be \(U(x) = -\alpha e^{-\beta x^{2}}\). Finally, normally a constant of integration would abound, but dealing with potential energy we can choose our zero level freely, and therefore we can set this constant to zero.

Key concepts

Understanding Conservative Forces

At the heart of many physical systems is the concept of a conservative force. These are forces that have the special property of conserving mechanical energy, regardless of the path taken by the object upon which they act. In other words, the work performed by a conservative force on an object moving between two points is the same, no matter what path is taken.

Examples of conservative forces include gravity and electrostatic forces. Notably, in the context of the provided exercise, knowing that a force is conservative allows us to deduce the existence of a potential energy function, which is a scalar field representing the potential energy per unit mass at every point in space. This potential energy function is especially significant because it is related to the force experienced by a particle at any position along its trajectory. It's crucial to recognize that the change in potential energy is negative to the work done by the conservative force.

The Nabla Operator in Physics

The nabla operator, denoted as abla, is a vector differential operator used extensively in physics. It enables the calculation of quantities like the gradient, divergence, and curl, which are central to the fields of classical mechanics, electromagnetism, and fluid dynamics.

In the simplest terms, the nabla operator is a symbolic representation that encapsulates the information of partial derivatives concerning the spatial variables. When applied to a scalar field like the potential energy function, it yields the gradient of that field, which tells us how the function changes over space. In the case of a one-dimensional problem like our textbook exercise, the nabla operator reduces to a simple derivative with respect to the single spatial variable.

Gradient in Physics

To gain deeper insights into conservative forces and potential energy, it's helpful to explore the gradient. In physics, the gradient represents the rate and direction of change in a scalar field. The gradient points in the direction of the greatest increase of the function, and its magnitude corresponds to the rate of increase.

In a physical sense, for a potential energy field, the gradient would indicate the direction in which the potential energy increases most rapidly. As seen in our exercise, the gradient also has a direct relationship with the force, given by abla U(x)abla U(x)—the force is the negative gradient of the potential energy function. This negative sign conveys that the force experienced by a particle will be in the direction that decreases its potential energy.

Integration in Physics

The process of integration in physics is a fundamental tool used to solve many problems, such as finding potential energies from forces. Integration can be thought of as summing infinitesimally small quantities to find a whole. It allows us to reconstruct a function (like potential energy) from its derivative (like force).

In our exercise, integration is employed to determine the potential energy function from the given force function. By integrating the negative of the force with respect to position, we are effectively accumulating the work done by the force over a certain distance, which corresponds to the change in potential energy. This integral gives us the potential energy function up to a constant, which can typically be set to zero at a point where the potential energy is arbitrarily defined to be zero.