# Chapter 9: Forces

## 9.3 Types of forces

In the classical model of physics, we describe many different forces depending on what things are interacting. A few of these forces have a distinct mathematical expression.

Weight is the force of gravity of the Earth pulling on an object. See section 9.8 for the full story. An object’s weight is given by the expression:

$F^G = mg \tag{9.5}$

The spring force is the force exerted by a spring that is stretched or compressed a distance x from its equilibrium position:

$F^\textit{sp} = kx \tag{9.6}$

This equation is known as Hooke’s Law after Robert Hooke. The $$k$$ is the same spring constant that we use to determine spring potential energy. We now see that the spring constant has units of N/m; the spring exerts some force for any given distance it is stretched or compressed from equilibrium. The spring force is always directed towards the equilibrium position of the spring. Hooke’s Law is a very powerful tool that is used to model many situations involving things which have a tendency to return to some equilibrium position.

Some of the forces we’ll use do not have a set equation to determine their strength. These forces will depend on everything else acting on a given object:

Normal force $$\left(F^N\right)$$ is the “force of contact” between two surfaces. It is always perpendicular to the interface of the two surfaces.

We can actually use Hooke’s law to better understand the normal force. A reliable way to model solids is to image the atoms as very small, hard spheres that are connected by springs. When something pushes down on this object, you can imagine those springs compressing. The animation below shows a simplified example:

Following Hooke’s law, the molecules then exert a force back up, towards their equilibrium position; this force is perpendicular to the surface itself. When taken over the many many many atoms that make up a solid, we have the effect which we call the normal force.

Friction $$\left(F^f\right)$$is the force between two surfaces that resists motion. It is always parallel to the interface of the two surfaces, and points in a direction to oppose motion. Ask yourself, “if there were no friction, which way would the object move?” The frictional force points in the opposite direction.

On a microscopic level, “smooth” surfaces are not smooth. When two surfaces rub against each other, the jagged “peaks and valleys” snag on each other; this resists the motion of the objects. This actually leads to two different kinds of friction: static and kinetic friction. Static friction is the frictional force that must be overcome to set an object into motion. In terms of microscopic “peaks and valleys,” overcoming static friction corresponds to the peaks and valleys becoming unstuck from one another. Static friction will increase in magnitude until it reaches some maximum; if the applied force exceeds the maximum static friction, the object begins to move.

Kinetic friction is the frictional force that resists the motion of an object once it is in motion. In general, kinetic friction is weaker than static friction. You may notice this yourself when trying to slide a heavy object across some surface: you have to push harder to get the object to even begin moving than to keep moving once in motion.

Tension $$\left(F^T\right)$$ is the force that a rope/wire/string/etc. exerts. It acts in the direction that the rope (or string, or wire…) is pulling. In general, you can think of tension as the force involved whenever you are using a tool to pull on something.

Sometimes we will refer to a force being “applied.” We use this somewhat generically, though it often is used to describe the force exerted by a push. We use the symbol $$\left(F^\textit{app}\right)$$ In the case of a push, this is technically a normal force! But it sometimes conceptually useful to just think of it on its own terms, as the force that you yourself are intentionally applying to an object.