Viscous friction (force) coefficient and examples

The viscous friction It arises when a solid object moves in the middle of a fluid - a gas or a liquid. It can be modeled as a force proportional to the negative of the speed of the object or to the square of it.

The use of one or the other model depends on certain conditions, such as the type of fluid in which the object is moving and whether or not it is very fast. The first model is known as linear resistance, and in it the magnitude of the viscous friction F_touch is given by:

F_touch = Γv

Here γ is the constant of proportionality or coefficient of viscous friction and v is the speed of the object. It is applicable to bodies moving at low speeds in fluids with a laminar regime.

In the second model, known as quadratic resistance or Rayleigh's law, the magnitude of the friction force is calculated according to:

F_touch = ½ ρ.A.C_d.v^two

Where ρ is the density of the fluid, A is the cross-sectional area of ​​the object, and C_d is the coefficient of aerodynamic drag.

The product ½ ρ.A.C_d  is an aerodynamic constant called D, whose SI units are kg / m, therefore:

F_touch = Dv^two

This model is more appropriate when the speed of the objects is medium or high, since the movement produces turbulence or eddies as it passes through the fluid..

A moving tennis ball and cars on the highway are examples of objects this model does quite well on..

The viscous force arises because the solid must push the layers of fluid apart in order to move through it. The existence of several models is due to the fact that this force depends on multiple factors, such as the viscosity of the fluid, the speed and shape of the object..

There are objects more aerodynamic than others and many are designed precisely so that the resistance of the medium reduces its speed to a minimum.

Article index

• 1 Examples of viscous friction
  • 1.1 Stones that fall into the water and pollen grains
  • 1.2 Swimmers and cyclists
  • 1.3 Paratroopers
  • 1.4 Automobiles
  • 1.5 Millikan's oil drop experiment
• 2 Exercises
  • 2.1 - Exercise 1
  • 2.2 - Exercise 2
• 3 References

Examples of viscous friction

Any person or object moving in a fluid necessarily experiences resistance from the environment, but these effects are often neglected for simple applications such as free fall..

In the statements of almost all free fall problems it is noted that the effects of air resistance are neglected. This is because air is a fairly "thin" fluid and that is why we expect that the friction it offers is not significant..

But there are other movements in which the viscous friction has a more decisive influence, let's see some examples:

Stones that fall into the water and pollen grains

-A rock that is dropped vertically into an oil-filled tube experiences a force that opposes its descent, thanks to the resistance of the fluid.

-Pollen grains are very small, so for them the air resistance is not negligible, because thanks to this force it is that they manage to stay afloat for a long time, causing seasonal allergies..

Swimmers and cyclists

-In the case of swimmers, they wear a cap and shave completely so that the resistance of the water does not reduce their speed..

-Like swimmers, time trial riders experience air resistance, consequently helmets have aerodynamic designs to improve efficiency.

Likewise, the cyclist's position within a competing group is relevant. The one who leads the march evidently receives the greatest air resistance, while for those who close the march, this is almost nil.

Paratroopers

-Once a skydiver opens the parachute, he is exposed to the viscous friction of the air, the most appropriate model being the one with the square of the speed. In this way it reduces its speed and as the friction opposes the fall, it reaches a constant limit value.

Automobiles

-For cars, the coefficient of aerodynamic drag, a constant that is determined experimentally, and the surface it presents against the wind, are the determining factors to reduce air resistance and reduce fuel consumption. That is why they are designed with sloping windshields.

Millikan's oil drop experiment

-In the Millikan oil drop experiment, physicist Robert Millikan studied the motion of oil drops in the midst of a uniform electric field, concluding that any electric charge is a multiple of the charge on the electron..

For this, it was necessary to know the radius of the drops, which could not be determined by direct measurement, given their small size. But in this case the viscous friction was significant and the drops ended up being stopped. This fact made it possible to determine the radius of the drops and subsequently their electrical charge..

Training

- Exercise 1

In the equation for the viscous friction force at low speed:

F_touch = Γv

a) What dimensions should the viscous friction coefficient γ have?

b) What are the units of γ in the International System of Units?

Solution to

Unlike the coefficients of static friction or kinetic friction, the viscous friction coefficient has dimensions, which must be:

Force / speed

Force has dimensions of mass x length / time^two, while those of speed are length / time. By denoting them as follows:

-Mass: M

-Length: L

-Time: T

The dimensions of the viscous coefficient of friction γ are:

[M.L / T^two] / [L / T] = [M.L.T / L.T^two] = M / T

Solution b

In SI, the units of γ are kg / s

- Exercise 2

Taking into account the resistance of the water, find an expression for the terminal speed of a metallic sphere that is dropped vertically into a tube filled with oil, in the cases:

a) Low speed

b) High speed

Solution to

The figure shows the free-body diagram, showing the two forces that act on the sphere: the weight downwards and the resistance of the fluid, proportional to the speed, upwards. Newton's second law for this motion states the following:

γv_t - mg = 0

Where V_t is the terminal speed, given by:

v_t = mg / γ

Solution b

If we assume medium to high speeds, the appropriate model is the one with the speed squared:

F_touch = ½ ρ.A.C_d.v^two

Then:

½ ρ.A.C_d.v^two - mg = 0

D.v^two - mg = 0

v = √ [mg / D]

In both situations, the greater the mass of the object, the greater its terminal velocity..

References

1. Serway, R., Jewett, J. (2008). Physics for Science and Engineering. Volume 1. 7th. Ed. Cengage Learning.
2. Sears, Zemansky. 2016. University Physics with Modern Physics. 14th. Ed. Volume 1.
3. Tipler, P. (2006) Physics for Science and Technology. 5th Ed. Volume 1. Editorial Reverté.
4. Tippens, P. 2011. Physics: Concepts and Applications. 7th Edition. Mcgraw hill
5. Sevilla University. Friction forces. Recovered from: laplace.us.es.