Coefficient of Static Friction Formula

The ratio of the force of friction acting between two bodies pressing against each other is the coefficient of static friction. 

So when the static force of friction between the surfaces in contact is maximum, also, before the movement, they commence to the Normal force ‘N,” we say that the ratio of μMAX and N is the coefficient of static friction. 

Since the surfaces in contact are static, we call the coefficient the coefficient of static friction. On this page, we will understand the force of the static friction equation and the kinetic friction force formula, followed by the coefficient of static friction formula, coefficient of kinetic friction formula, respectively.  

What is Friction?

Friction is the resistive force that acts between the relative motion of the material elements sliding against each other, solid surfaces, and fluid layers.

Now, let’s understand the force of static friction equation:

Static Force Friction Formula

If ‘f’ is the friction force, ‘N’ is the force acting normal or perpendicular to the surface, and ‘μ’ is the coefficient of friction, then static friction formula is given by:

                                f  =  μN….(1)

Or,

                               μ  =  F/N….(2)

In the above static friction force formula (2), we see that μ is a unitless quality.

Coefficient of Static Friction

Coefficient of static friction is the maximum ratio of applied force to normal force with no movement.” 

You know that frictional force is a force that opposes the motion of an object.

Say, you’re climbing an icy mountain. The sand deposited on the ice, helps you climb smoothly without having chances of slipping.  Here, sand works as a frictional force or the savior of your life.

If you keenly observe something here, once you apply force to move from the rest position, there exists a point up to which your body resists moving. Now, to move, the applied force should be greater to overcome this resistance. 

So, this maximum resistive force offered by sand against the slippery melted icy surface to continue your state of motion is called the Coefficient Of Static Friction.

Static Friction Formula

The coefficient of static friction mainly relies on frictional force. Following this, the frictional force magnitude depends on the following two factors:

Firstly, how heavy is the material/object.

Secondly, how rough are the surfaces in contact?

Therefore, the roughness of the surface is most frequently measured by a parameter called the Coefficient Of Static Friction. Where we express this factor mathematically as the ratio of applied force to the normal reaction. It is given by:

                          μS   =  \[\frac{N_{s}^{\rightarrow}}{F_{n}^{\rightarrow}}\]

So, the static friction coefficient is represented by μS. It is a unitless quantity because the Fs→ and Fn→  are the forces having the same unit that got cancelled. Also, we can say that ‘μS’ is a dimensionless quantity, 

Now, let’s have a look at the coefficient of static friction example to understand the static and kinetic friction formula:

Coefficient of Static Friction Example

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In the above arrangement, you see that a heavy wooden box is placed on the floor. 

Now, if you try to push the box away from you because of the presence of friction the box says ‘No No I’m not going in that direction!’ The reason for this is the roughness of the surfaces of the box and the floor. 

Here, on observing the point of contact between the box and the floor with higher resolution, we notice microscopic deformations. These tiny deformations present on the surface are the reasons for the resistance to the motion. 

Therefore, the amount of resistance to the motion relates directly to the force of friction. To understand all about this, it is significant to know about the forces that act on the box along with its directions, which is the free body diagram.

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As a virtue of the mass of the box, the force of gravity (Fg→ ), which is equal to mg N will be acting downwards.

The force pushing the box upwards, i.e., away from the gravity is the normal force (FN→ ), which we know that FN→   =   mg N.

The force you are applying on the object to move/push is the applied force, which is  FS→ 

A resistive force exerted by the object to oppose the movement of the box is the Frictional force  FAPPLIED→.

Calculating Static Friction

For calculating static friction, we use the below formula:

                        μS   =   \[\frac{N_{s}^{\rightarrow}}{F_{n}^{\rightarrow}}\]

                        FS→  =   FN→. μS      (This is the required Static Friction Equation)

Therefore, the formula for calculating static friction is given as;

Coefficient of Static Friction Value

You notice that the moment you start pushing the box away from you as shown in diagram (1). 

Initially, it remains static, you push it and it pushes you back. However, on applying a strong force, the box starts moving. 

At this point, there commences a small interval between the moment you started pushing the box and the movement the box started moving, that is where the coefficient of static friction comes into the display. From here, we get the coefficient of static friction value, and this value plays a crucial role in many real-life applications.

There’s another concept along with the static friction, which is the kinetic friction formula, now let’s understand this followed by the coefficient of kinetic friction formula and then static and kinetic friction formula.

Kinetic Friction Formula

Similar to static friction, kinetic friction acts between two surfaces in contact with each other; however, these surfaces remain in motion (not static). Also, the magnitude of kinetic friction relies on the kinetic friction between the two surfaces.

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The kinetic friction force formula is given as:

                            FK→  =     μK. FN→   

Here, the coefficient of kinetic friction formula is:

                             μK   =   \[\frac{F_{k}^{\rightarrow}}{F_{n}^{\rightarrow}}\]               

Now, let’s understand the static and kinetic friction formula:

Static and Kinetic Friction Formula

The static and kinetic formula is as follows:

We know that   μS = \[\frac{N_{s}^{\rightarrow}}{F_{n}^{\rightarrow}}\] and μK   =   \[\frac{F_{k}^{\rightarrow}}{F_{n}^{\rightarrow}}\] , we must note that relation between the coefficient of static friction and the coefficient of kinetic friction is:            

The below graph displays the kinetic and the static region of the object possessing these frictional forces:

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Conclusion

We see that be it a static friction force or kinetic friction force, the force of friction between two static/moving surfaces in contact with each other always equals the product of the normal force multiplied by the coefficient of static/kinetic friction.