# Differential Equations Applications - Mass on a Spring

Differential equations are used for various applications. When a mass on a vertical spring is released, the governing equation is:

$$\sum F = ma = m \cfrac{dx^2}{dt^2} = F_g + F_s + F_d + F(t)$$

$$m \cfrac{dx^2}{dt^2} = mg - k(s + x) -\beta \cfrac{dx}{dt} + F(t)$$

Where $$k$$ is the spring constant, $$s$$ is the distance the spring moves when the mass is applied and $$\beta$$ is a damping constant. When the spring is at the equilibrium position $$mg - ks = 0$$.

There are different scenarios which will lead us to use different strategies to solve the DEs.

Free motion occurs when there is no external force $$F(t)$$.

Undamped motion occurs when $$\beta = 0$$.

## Free Undamped Motion

$$m \cfrac{dx^2}{dt^2} = mg - k(s + x) = mg -ks -kx$$

$$m \cfrac{dx^2}{dt^2} + kx = 0$$

This is a homogeneous linear DE with constant coefficients that can be solved using by solving a characteristic equation.

$$m r^2 + k = 0$$

$$r = \sqrt{-\frac{k}{m}} = \sqrt{\frac{k}{m}} i$$

The general solution is:

$$x(t) = c_1 \sin {\sqrt{\frac{k}{m}} t} + c_1 \cos {\sqrt{\frac{k}{m}} t}$$

It is common to let $$\omega ^2 = \frac{k}{m}$$:

$$x(t) = c_1 \sin {\omega t} + c_1 \cos {\omega t}$$

## Free Damped Motion

$$m \cfrac{dx^2}{dt^2} + \beta \cfrac{dx}{dt} + kx = 0$$
 Roots Scenario Determinant Solution 2 Real Roots Overdamping $$\beta ^2 - 4mk > 0$$ $$x (t) = e^{\frac{\beta}{2m}t} (c_1 e^{\sqrt{(\frac{\beta}{2m})^2 -\frac{k}{m}t}} + c_2 e^{-\sqrt{(\frac{\beta}{2m})^2 -\frac{k}{m}t}})$$ Repeated Real Roots Critical Damping $$\beta ^2 - 4mk = 0$$ $$x (t) = e^{\frac{\beta}{2m}t} (c_1 + c_2 t)$$ Complex Roots Underdamping $$\beta ^2 - 4mk < 0$$ $$x (t) = e^{\frac{\beta}{2m}t} (c_1 \cos {\sqrt{\frac{k}{m}t - (\frac{\beta}{2m})^2}} + c_2 \sin {\sqrt{\frac{k}{m}t - (\frac{\beta}{2m})^2 }})$$