Question

Discussion problems.

21.Discuss why the damping term in equation(3) is written as

$\beta \left|\frac{dx}{dt}\right|\frac{dx}{dt}$ instead of $\beta {\left(\frac{dx}{dt}\right)}^2$

Short answer

It was necessary to write $\left|\frac{dx}{dt}\right|\frac{dx}{dt}$ instead of $\frac{d^{2}x}{d{t}^2}$ because the velocity (and then the damping force, too) is changing the sign.

Step-by-step solution

Definition of non-linear differential equation

The non-linear differential equation

$m\frac{d^{2}x}{d{t}^2}+\left|\frac{dx}{dt}\right|\frac{dx}{dt}+kx=0$

represents a model of a free spring (or) mass system where damping force is proportional to the square of velocity.

It was necessary to write$\left|\frac{dx}{dt}\right|\frac{dx}{dt}$ instead of $\frac{d^{2}x}{d{t}^2}$ because,

If we denote the equilibrium position of a spring with the mass attached on it and if we set it in motion, it will oscillate around it.

Representation of damping term 

The velocity will have both the positive and the negative sign.

Since the damping force is proportional to it, it will also have to change the sign.

$\frac{d^{2}x}{d{t}^2}$ is always positive.

That is why we had to write $\left|\frac{dx}{dt}\right|\frac{dx}{dt}$

Therefore, It was necessary to write$\left|\frac{dx}{dt}\right|\frac{dx}{dt}$ instead of $\frac{d^{2}x}{d{t}^2}$ because the velocity (and then the damping force, too) is changing the sign.