# Dynamic or kinetic friction: coefficient, examples, exercises

The **dynamic friction ****or kinetic** is one that occurs between two bodies in contact when the surface of one of them moves to the surface of the other. For example, in a box that slides downhill, friction is of the dynamic type and is distributed over the contact surface of the block.

The slope must be large enough so that the tangential component of the weight is equal to or greater than the friction force; otherwise, the descending block would end up stopping.

The friction force is extremely important in everyday life, as it allows people, animals and vehicles to move around. On a frictionless surface, such as a frozen lake, it is not possible to initiate motion.

Friction also allows our cars to stop when they are in motion.

When applying the brakes, the brake pads are pressed against the wheel discs and, thanks to dynamic friction, stop their rotation. But it’s not enough to have good brakes, it’s necessary to have enough friction force between the tires and the floor, because, finally, that’s the force we depend on to get the car to stop.

Humanity has learned to handle friction for its benefit. So he started using friction between two pieces of dry wood to make a fire.

Nature has also learned to handle friction to its advantage. For example, the synovial membranes that cover the bones of joints are one of the surfaces with the lowest coefficient of friction in existence.

**Dynamic friction coefficient**

The first to systematically study the motion of a block sliding on a flat surface was Leonardo da Vinci, but his studies went unnoticed.

It wasn’t until the 17th century that French physicist Guillaume Amontons rediscovered the laws of friction:

**Dynamic Friction Laws**

1.- The friction force present in a block that slides on a flat surface always opposes the direction of movement.

2.- The magnitude of the dynamic frictional force is proportional to the clamping force or normal force between the surfaces of the block and the support plane.

3.- The proportional constant is the friction coefficient, static μ _{and} in case of non-slip and dynamic μ _{d} when there is. The friction coefficient depends on the materials of the surfaces in contact and the roughness state.

4.- The friction force is independent of the apparent contact area.

5.- Once the movement of one surface in relation to the other is started, the friction force is constant and does not depend on the relative speed between the surfaces.

In the case of non-slip, static friction is applied whose force is less than or equal to the coefficient of static friction multiplied by the normal.

The last property was the result of the contribution of French physicist Charles Augustin de Coulomb, best known for his famous law of force between point electrical charges.

These observations lead us to the mathematical model of the dynamic friction force **F** :

**F** = μ _{d }**N**

Where µ _{d} is the dynamic coefficient of friction and **N** is the normal force.

**How to determine the coefficient of dynamic friction?**

The coefficient of dynamic friction between two surfaces is experimentally determined. Its value depends not only on the materials of both surfaces, but also on the roughness or polish they have, as well as their cleanliness.

One way to determine this is to drive and slide a box of known mass onto a horizontal surface.

If the speed at the time of driving is known and the distance traveled from the moment to the stop is measured, it is possible to know the acceleration of braking due to dynamic friction.

**To experiment **

In this experiment, the initial velocity *v* and the distance *d* are measured so that the acceleration of the braking is:

*a = – v ^{2} /2d*

The force diagram is shown in Figure 2. The magnitude of the weight is the mass m of the block multiplied by the acceleration due to gravity g and, as you know, the weight always points vertically downward.

**N** is the normal force due to upward thrust from the bearing surface and is always perpendicular (or normal) to the plane. Normal exists as long as the surfaces are in contact and ceases as soon as the surfaces separate.

Force **F** represents the dynamic friction force. It is actually distributed on the bottom surface of the block, but we can represent it as a single force **F** applied to the center of the block.

As there is vertical balance, the magnitude of normal **N** is equal to that of the mg weight:

N = mg

In the horizontal direction, the friction force causes the block of mass m to decelerate according to Newton’s second law:

-F = ma

The friction force **F** points to the left, so its horizontal component is negative, m is the block mass and is the braking acceleration.

Previously, *a = – v ^{2} / 2d* had been obtained and also the dynamic friction model indicates that:

F = μ d N

Substituting in the previous equation, we have:

-μ _{d} N = *– v *^{2}* /2d*

Taking into account that N = mg, the coefficient of dynamic friction can now be cleared:

μ _{d} = *v ^{2}*

*/ (2d mg)*

**Table of friction coefficient of some materials**

The table below shows the coefficients of static and dynamic friction for various materials. It should be noted that systematically the static friction coefficient is always greater than the dynamic friction coefficient.

**Exercises**

**– Exercise 1**

A 2 kg block of mass is pushed onto a horizontal floor and released. When released, a speed of 1.5 m/s is recorded. From that moment until the blocking is broken by dynamic friction, 3 m are traveled. Determine the coefficient of kinetic friction.

**Solution**

According to the formula obtained in the example in the previous section, the coefficient of dynamic (or kinetic) friction is:

μ _{d} = *v ^{2}*

*/ (2d mg) =*

*1.5*

*2*

*/ (2x3x2 x9.8) = 0.019*.

**– Exercise 2**

Knowing that the block in figure 1 descends with constant velocity, that the mass of the block is 1 kg and that the slope of the plane is 30º, determine:

a) The value of the dynamic friction force

b) The coefficient of dynamic friction between the block and the plane.

**Solution**

Figure 4 shows the equation of motion (Newton’s second law) for the problem of a block going down a slope with coefficient of friction µ _{d} and slope α (see the force diagram in Figure 1)

In our exercise, we are told that the block descends at a constant speed; therefore, it descends with acceleration a = 0. Thereafter, the friction force is such that it equals the tangential component of the weight: F = mg Sen (α).

In our case, m = 1 kg and α = 30º, so the friction force F has a value of 4.9N.

On the other hand, the normal force N is equal and opposite to the perpendicular component of weight: N = mg Cos (α) = 8.48N.

As a result, the coefficient of dynamic friction is:

μ _{d} = F / N = 4.9N / 8.48N = 0.57