ENGR201 Laplace deck tutorial Q7, water tank struck by a gust (impulse) (ENGR5001)

ENGR201 Laplace deck tutorial Q7, water tank struck by a gust (impulse)TutorialOld spec ENGR2012:212 min

ENGR201 (old spec) Laplace transforms deck, Q7 of tutorial sheet, water tank (pages 28-29)

A water tank of mass $m = 1$ is supported by a column with stiffness constant $k = 5$ and damping constant $c = 2$ (a mass-spring-damper in horizontal motion, displacement $x(t)$ ). At the instant $t = \pi$ the tank is struck by a gust of wind giving an impulse $p = 10$ . Also $x(0) = 0$ and $x'(0) = 0$ .

Formulas you may need

$\mathcal{L}[\delta(t - c)] = e^{-cs}$ ; $\mathcal{L}[x''] = s^2X - sx(0) - x'(0)$ (learn this)
Shift: $\mathcal{L}^{-1}[e^{-cs}F(s)] = f(t - c)u(t - c)$ ; $\mathcal{L}^{-1}\left[\dfrac{b}{(s - a)^2 + b^2}\right] = e^{at}\sin bt$ (learn this)

(a)
Write the differential equation governing the motion of the tank.
(b)
Find the Laplace transform $X(s)$ of the solution.
(c)
Find $x(t)$ , and evaluate it at $t = \pi + 0.5$ .
x(pi + 0.5) =