Static Friction

Static Friction: Definition

If we have a box on the floor and we try to push it with some force, the box may just stand still and not move. This is due to the static friction force. The static frictional force occurs when the object and surface are at rest relative to each other. There is no motion of one relative to the other. Due to static friction, an object will stay still on a surface without slipping. The floor exerts a static friction force that is equal in magnitude and opposite in direction to the applied force until the force applied is greater than the maximum static frictional force. The static friction force can be shown as \(\overset\rightharpoonup{f_{\mathrm s}}\).

Fig. 2 - A box initially at rest experiencing an applied force \(\overset\rightharpoonup F\) to overcome the force of static friction.

In the figure above, we see a box that is initially at rest. The normal force \(\overset\rightharpoonup N\) is upwards and is exerted on the box by the floor, while the weight \(\overset\rightharpoonup W\) acts downward. Because the box is at equilibrium, the magnitudes of the normal force and weight are equal.

Then a force \(\overset\rightharpoonup F\) is applied on the box pulling it to the right, over time the magnitude of this applied force is gradually increased. The box will stay at rest for some time. This is because even if the force is increasing, the force of static friction is increasing as well to balance the magnitude with force \(\overset\rightharpoonup F\).

The force needed to move the box is equal and opposite to the maximum static friction force. The maximum value for the static friction force is shown as \({(f_{\mathrm s})}_\max\) and it is proportional to the magnitude of the normal force. The proportionality factor is shown as \(\mu_{\mathrm s}\) and is called the coefficient of static friction. Once the maximum static friction force is overcome, the box will begin to slide to the right.

Static Friction: Formula

The real value of static friction might range from zero (when there is no force exerted on the object) to a maximum value, which is mathematically represented as

$$f_s\leq{(f_s)}_{max}=\mu_sN.$$

According to the mathematical representation, the static friction relationship is between magnitudes, as the static friction magnitude is proportional to both the magnitude of the normal force and the roughness between the two surfaces in contact. Therefore, directions do not matter like in vectorial relationships.

So, what does this formula mean?

• When the exerted force \(\overset\rightharpoonup F\) is at a value at which the motion starts, the formula can be used to calculate the magnitude of the static friction force.
• If the force \(\overset\rightharpoonup F\) is not causing the object to move, then the magnitude of static friction force should be less than \(\mu_sN\). In this case, the magnitude of static friction force will simply be equal to the magnitude of the applied force \(\overset\rightharpoonup F\).

Limiting Friction

We mentioned that there is a limit where the force \(\overrightarrow F\) is greater than the maximum value of the static friction force. If the limit is surpassed, the box begins to move. When the static friction force is at a maximum and the motion is about to start, the frictional force is called the limiting friction. When motion starts, the object will no longer experience static friction. Instead, the object experiences the force of kinetic friction.

Fig. 3 - Diagram to show how both the forces of static friction and kinetic friction change with an increasing applied force.

If we look at the above graph we see that with an increasing applied force \(\overset\rightharpoonup F\), the magnitude of the static friction force begins to increase as well until it reaches a peak where the magnitude is that of the limiting friction. Afterwards, the object begins to move, and the static friction force is no longer effective, but the kinetic friction force is.

Static Friction Examples

Now let's investigate some examples involving the static friction force.

Static Friction vs Kinetic Friction

So, what is the difference between static friction and kinetic friction?

The differences between the forces of static friction and kinetic friction can be summarised as:

1. Static friction is only applied while the object is stationary, while kinetic friction force is applied when the object is moving. (A way to remember the key difference between them is to think critically about their definitions. While static means lacking in movement, kinetic means relating to or resulting from motion.)
2. The maximum value of the static friction force is always greater than the magnitude of the kinetic friction force because it is always easier for an object to maintain motion than it is to begin motion from rest.
3. The value of the force of kinetic friction across a dry surface is essentially constant, no matter the speed of the object. Whereas the magnitude of static friction increases with the applied force until it reaches the limiting friction.