# Differential equation of frictional force

• Dazed&Confused
In summary, the conversation discusses a question from a classical mechanics past paper about a particle with horizontal identical springs attached and placed on a table with friction. The conversation also includes a proposed solution to the equation of motion and a discussion about the need for a sign in the friction term to accurately describe the motion of the particle. It is ultimately concluded that the particle should not stop at the equilibrium position, as it would result in a net force on the particle.
Dazed&Confused
A question from a classical mechanics past paper described a particle of mass
##m## that had a pair of horizontal identical springs of spring constant ##k## attached on either side and that the mass is free to move horizontally. The mass is also placed on a table that gives rise to an additional frictional force; the coefficient of sliding friction between the mass and the table is ## \mu ##.

You have to 'verify' that the following is a solution of the equation of motion:

$$x_B(t) = B \sin( \omega_B t) + C( \cos( \omega_B t) - 1).$$

In my opinion this is wrong. The solution clearly should decay in time. It seems that the differential equation they wanted was of the form

$$x''(t) = -2kx(t) - \mu mg.$$

I think it should be

$$x''(t) = -2kx(t) - \mu mg \text{Sign}(x'(t)),$$

where ##\text{Sign}(x)## is defined to be 1 if ##x > 0## and -1 if ## x < 0. ##

Is there something wrong with my thinking? Keep in mind that Mathematica's numerical solution does show a particle approximately exhibiting SHM motion, but also decaying with time, however it does not ultimately stop moving at the equilibrium position.

You definitely need a sign in the friction term or it only describes a constant force acting in a determined direction. This is equivalent to a gravitational force on a vertical spring with a mass attached and does nothing but translate the solution to be oscillations around the new equilibrium position.

Dazed&Confused said:
however it does not ultimately stop moving at the equilibrium position.

Why do you think it would stop moving at the equilibrium position? (I assume you here mean the position where the spring is at its lowest potential energy.)

Well if it stopped at any other position there would be a net force on the particle due to the springs, since I haven't added a static friction term.

## What is a differential equation of frictional force?

A differential equation of frictional force is a mathematical expression that represents the relationship between the magnitude of frictional force and the relative velocity between two surfaces in contact. It takes into account factors such as surface roughness, normal force, and material properties.

## How is a differential equation of frictional force derived?

A differential equation of frictional force is derived using Newton's second law of motion, which states that the force on an object is equal to its mass multiplied by its acceleration. By considering the forces acting on an object in contact with a surface, and using the definition of frictional force, a differential equation can be formed.

## What are some applications of differential equations of frictional force?

Differential equations of frictional force are commonly used in engineering and physics to model and predict the behavior of objects in contact with each other. They are particularly useful in designing and optimizing systems that involve sliding or rolling motion, such as car brakes and bearings.

## What are the limitations of differential equations of frictional force?

One limitation of differential equations of frictional force is that they assume the surfaces in contact are smooth and rigid. In reality, surfaces are often rough and can deform under pressure, leading to more complex frictional behaviors. Additionally, these equations do not take into account other factors such as temperature and humidity, which can also affect friction.

## How can differential equations of frictional force be solved?

Differential equations of frictional force can be solved using various mathematical techniques, such as separation of variables, substitution, and numerical methods. The specific method used depends on the complexity of the equation and the desired level of accuracy.

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