# Applications of Second Order Differential Equations

Hanging Spring.

A 10 kilogram mass suspended from the end of a vertical spring stretches the spring $$\frac{49}{90}$$ metres. At time t = 0, the mass is started in motion from the equilibrium position with an initial velocity of $$1$$ $$m/s$$ in the upward direction. At the same time, a constant downward force of $$360$$ Newtons is applied to the system.

Assume that air resistance is equal to $$60$$ times the instantaneous velocity and that the acceleration due to gravity is $$g = 9.8$$ m/s^2.

(a) Determine the spring constant.

(b) Show that the equation of motion is

$$\ddot{x} + 6 \dot{x} + 18x = 36$$

where $$x(t)$$ is the displacement of the mass below the equilibrium position at time $$t$$ . In your answer include a diagram of all forces acting on the mass.

(c) Find the position of the mass at any time. Would you describe the motion as overdamped, underdamped or critically damped?

I'm struggling with part (c) and any help would be appreciated.

Solving for $$\lambda$$ I know that

$$X_p =A\cdot t\cdot e^{-3t}\cdot\cos(3t)+B\cdot t\cdot e^{-3t}\cdot\sin(3t)$$

since without $$t$$, using the equation gives a result of zero,

using

$$\ddot{x} + 6 \dot{x} + 18x = 36,$$

with $$X_p$$ I get

$$6=e^{-3t} (b \cos(3t)-a \sin(3t))$$

$$x(t) = −2 e^{−3t} \cos (3t) −\frac73 e^{−3t}\sin (3t) + 2$$
where does this $$+2$$ come from, and how is the equation used without multiplying by t if it equals to zero? Sorry if this question is unclear and if any clarification is needed please ask.
• Where is $\lambda$ defined in your question? – BLAZE Sep 26 '15 at 10:56