Force vs. Position Graph

Force vs. Position Relationship

Before we dive right into all the fun with graphs (I can just sense your excitement), we have to go over a key definition: directly proportional relationships.

This definition will come in handy as you continue reading, especially when you get to all the talk about spring constants. When you get there, think about what it means for force and position to have a directly proportional relationship. If \(k\) is the slope of our force vs. position graph and also a constant, what does that mean for the relationship between force and position?

Force vs. Position Interpretation

One of the essential skills in physics to develop is finding relationships. Physics is all about how one thing is related to another. That is how we get all of our equations; it is why modeling relationships through graphs is meaningful. Becoming relationship-minded and asking how the concepts you learn relate to each other will accelerate your understanding of physics.

Area under a Force vs. Position Graph

The area under a force vs. position graph is equal to the work done by the force on whatever object is getting displaced. Recall the equation

$$W = F \Delta x$$

that describes the work done that you learned in AP Physics 1.

Notice how work is simply the product of force and position. Recognizing this relationship makes understanding the area of a force vs. position graph easier. The area under the curve is equal to the work because the integral of the graph will relate the force to the object's position. Therefore, a more advanced equation for work can be obtained by applying integration like so:

$$W = \int_{\vec a}^{\vec b} \vec{F}\cdot \mathrm{d}\vec{r}$$

where \(W\) is the work done, \(\vec a\) and \(\vec b\) are your initial and final positions, and \(\vec F\) is the force as a function of position \(\vec r\).

Fig. 1 - Here, the area under the curve of a force vs. position graph is shaded and labeled as work.

Notice that the work equation involves evaluating the dot product of two vectors.

Expressed mathematically, a dot product is

$$\vec{A} \cdot \vec{B} = A B \cos{\theta},$$

where \(\vec A\) and \(\vec B\) are our two vectors, \(A\) and \(B\) are their magnitudes, and \(\theta\) is the angle between them.

Force vs. Position Slope

Remember how we said that physics is all about finding relationships? Let's see if we can find a relationship between force and position that will explain the slope of a force vs. position graph.

Force vs. Position Slope Meaning

The slope is equal to the rise over the run; therefore, the slope of our graph would look something like this:

$$\frac{F}{\Delta x}\\\mathrm{.}$$

Does this equation ring any bells? It is the formula for the spring constant \(k\) that shows up in Hooke's Law.

Written mathematically, Hooke's Law looks something like this

$$F_\text{spring}=-kx,$$

where \(F_\text{spring}\) is the magnitude of the force of the spring and \(x\) is the spring's distance from equilibrium.

This means that a force vs. position graph that was referring to the displacement's effect on the force of a spring would have a spring constant that could be calculated using the formula

$$k = \frac{F}{\Delta x}\\\mathrm{.}$$

Fig. 3 - Notice that the variable \(k\) is being multiplied by \(x\) to show that \(k\) is the slope. Since \(k\) is a constant value, the force and the displacement above have a directly proportional relationship.

The slope of a force vs. position graph is constant, as seen in Fig. 2 above when we refer to springs and Hooke's Law: in that specific case, the slope of our force-displacement graph is constant and equals the spring constant \(k\). However, when generalizing the slope of a force vs. displacement graph, the slope does not have to be constant. For example, \(F(x)\) could, perhaps, be a variable force described by a higher-order equation, meaning that the slope of our graph could follow a more parabolic or cubic trajectory. Therefore, in a general sense, we have to say that the slope of a force vs. position graph is its derivative, which can have applications for many other physical scenarios: the spring constant \(k\) is only one example in case the force is directly proportional to the displacement.

Therefore, we can rewrite our equation for a 'variable spring constant' \(k_\text{variable}\) in a new light, involving our knowledge of derivatives:

$$k_\text{variable}=\frac{\mathrm{d}}{\mathrm{d}x}F(x).$$

A Closer Look at \(k\)

For ideal springs, \(k\) is not variable and is a constant depending on the inherent characteristics of the spring.

Fig. 4 - As the spring expands or contracts (displaces some amount \(x\) from equilibrium), the force exerted changes; the constant \(k\) quantifies the stretchiness of the spring.

The graph in Fig. 4 takes a closer look at the meaning of the spring constant \(k\). As the spring displaces some amount \(x\) from equilibrium, the restoring force tries to pull it back (represented by the green arrow \(F_\text{s}\)). The force exerted on the spring, causing it to displace is represented by the purple arrow \(F\). The spring constant \(k\) is the stretchiness of the spring, it quantifies how difficult it is for our two forces to displace the spring, or in other words, to change \(x\).

Force vs. Position Slope Units

Through a similar process, the units for the slope of a force vs. position graph can be found. Start with our work equation,

$$\frac{F}{\Delta x}\\\mathrm{,}$$

and recognize that the units for force are newtons, and for position, meters:

$$\mathrm{\frac{N}{m}\\}\mathrm{.}$$

These units make a lot of logical sense. Seeing as how \(k\) is the spring constant, the fact that its units are newtons over meters shows that it can be found by doing the force per unit length. This gives the stretchiness of the spring. Recall that newtons are given as

$$\mathrm{N=\frac{m\,kg}{s^2}\\}\mathrm{,}$$

therefore, we can relate our initial understanding of the units, force over meters, to a new understanding:

$$\mathrm{\frac{N}{m}\\=\frac{kg}{s^2}\\}\mathrm{.}$$

Going even further allows us to see how the spring constant relates to surface tension. Surface tension is found by doing the force over a length, and it has units of kilograms over seconds squared, just like the spring constant.

Force vs. Position Graph and Velocity

Force vs. position graphs give us enough information to find the velocity of an object. By calculating the work from the graph by finding the area under the curve, one can find the velocity using the Work-Energy Theorem.

So now, when your mom tells you to take out the trash, tell her that you could do more work by simply lifting a pencil (that is if you fail to actually move the trash can).