Review of Force and Potential Energy

In the previous chapter, we developed a foundation for how we expect atoms to interact as they near each other. In this chapter, we will explore a mathematical model of atomic bonding and see how it can be used in computer simulation. Broadly, the learning goals are to:

  1. Construct and utilize mathematical models of bonding to make predictions
  2. Understand the fundamentals of a computational modeling technique known as molecular dynamics
  3. Explore cross-cutting physical principles using mental, mathematical, and computational models.

Before getting there, we must review the physical definitions and concepts of force and potential energy to better understand the strength of the interactions between atoms.

Let's work first with a familiar system: an idealized spring, shown in Figure 4.3.1. The mechanical behavior of this spring is governed by Hooke's Law. This spring is oriented in the $x$-direction, and one end is positioned at $x = 0$ when in equilibrium. When displaced in the positive $x$-direction, there is a restoring force $F$ acting to pull the spring's end back to $x=0$ that is linearly proportional to the distance, $x$, the end of the spring was displaced. Similarly, when compressed along the $-x$-direction, there is a restorative force acting in the $+x$ direction.

Illustration of a spring and its restoring force when displaced from equilibrium.

Figure 4.3.1 Illustration of a spring and its restoring force when displaced from equilibrium.

This relationship between force and position can be written mathematically as the (perhaps familiar?) Hooke's Law:

$$F(x) = -k x \tag{4.3.1}$$

where $k$ is the spring constant—a measure of how stiff the spring is, $F$ is the force, and $x$ is the distance the spring has been displaced with respect to the equilibrium position at $x = 0$. A higher $k$ means a larger restoring force when the spring is displaced. We demonstrate Hooke's law (Eq. 4.3.1) graphically in Figure 4.3.2. There, a red (solid) line shows how the force $F$ changes as a function of $x$ with a constant and arbitrary value of $k$.

Note that $F$ is positive when the spring is compressed ($x< 0$), indicating a restorative force in the $+x$ direction. This is the convention we will use in this text: the sign of the force indicates the direction in which the force is acting. In compression, one think of this as a "repulsion," as the spring would push your hand away. $

F$ is negative when the spring is in tension ($x > 0$), indicating a restorative force in the $-x$ direction. One might think of this as an "attraction," as the spring would act to pull on your hand. We'll use this idea of positive force as repulsion and negative force as attraction in the following sections.

Figure 4.3.2 The force (Eq. 4.3.1) and potential energy of an ideal spring as a function of displacement from equilibrium.

As we extend or compress the spring, we are storing mechanical energy in the system. We relate the force $F(x)$ and the potential energy $U(x)$ through the following relationship:

$$F(x) = - \frac{dU(x)}{dx} \tag{4.3.2}$$

That is, the force at position $x$ is the negative derivative of the potential energy at that position. In Exercise 4.3.1, we plot $U(x)$ as the purple dashed line. Intuitively, you can think of the potential function as a landscape that a ball is rolling around in. The ball feels a force that causes it to roll downhill. The magnitude of that force is proportional to the slope at the the position it is located. When potential energy is at a local minimum, the force is zero (this defines equilibrium; all forces are balanced). The further the ball is displaced from this local minimum, the higher its energy.

Let's explore the spring model in Exercise 4.3.1.

Exercise 4.3.1: Spring Model Exploration
Not Currently Assigned

Let's build intuition using the spring model.

Communicating clearly with mathematics is important in science and engineering. The best way to communicate with mathematics is to use the correct mathematical symbols and notation. The best way to use the correct mathematical symbols and notation is to use $\LaTeX$.

You don't need to know much about $\LaTeX$ to use it here, but if you spend a few minutes becoming familiar with it, it will pay dividends in the future. I used Overleaf, which has a few short tutorials that will teach you a lot in about 30 minutes. Here, feel free to use palette-based editor like this one.

This exercise should take around 15 minutes to complete.

  1. As the spring stretches, does it get harder or easier to stretch it further? Think about this in terms of the force necessary to stretch a spring to 1 m compared to 5 m.

  2. Look at the graph in Figure 4.3.2 (no need to compute). Does it take more energy to pull the spring from $x=0$ to $x=1$ or from $x=1$ to $x=2$?

    Show why, mathematically, using Eq. 4.3.1 and Eq. 4.3.2.

  3. Compute an expression for the potential energy of the spring as a function of $x$, showing clearly that it is parabolic.

    It's best to show your math, here, using this editor and enclosing your syntax in dollar signs: i.e., $F(x) = -kx$ will show $F(x) = -kx$.

  4. Assume that the spring constant $k = 1\,\mathrm{N/m}$. What is the potential energy of the spring when stretched to $1 \,\mathrm{m}$? How about $2 \,\mathrm{m}$? Make sure your answers agree with the qualitative answers from the previous parts. (i.e., define the potential energy to be zero when the spring is at equilibrium).

    We're asking you to mathematically calculation what you explained in Part 2.