Question

(5 points) A spring is suspended vertically from a fixed support. The spring has spring constant k 40 N m-1.An object of mass m- kg is attached to the bottom of the spring. The subject is subject to damping with damping constant β N m-1 s. Let y(t) be the displacement in metres at the end of the spring below its equilibrium position, at time t seconds. (a) Give a value of B which would result in underdamping. B4 Give your answer as an exact value or as a decimal to at least 3 decimal places. (b) Which of the following plots of y vs. t shows underdamped motion? (c) Give the value of β which would result in critical damping, β- 4*sqrt(2) Give your answer as an exact value or as a decimal to at least 3 decimal places. (d) Suppose that an external force of F(t) Newton directed downwards is applied to the spring. With β = 0, for which function(s) F from the list below would resonance occur? Select one or more of the following A. 5 cos(-5V8t) B. sin(5v81) -t C. 4 cos(V81) + e D. cos(V81) E.e-r sin(v/81) F 2 sin(v8) G. e-' sin(5v8t) (e) Which of the following plots of y vs. t show resonance? Note: In order to get credit for this problem all answers must be correct.

Answer 1

Solution:

a)

For under damping to happen

$\beta^2-4mk<0=>\beta^2-4*\frac{1}{5}*40=>\beta^2>32=>\beta<4\sqrt{2}$

This means under damping would happen when 0< 4V2 so let us assume

4

$\text{b)}$

The DE would be

general solution would be

$x(t)=e^{-10t}\left ( c_1\cos (10t)+c_2\sin (10t) \right )$

Assuming c1, c2 to be 1 and 2 respectively

-10t r(t)t (cos(10t) 2 sin(10t) COS

$v(t)=x'(t)=-\mathrm{e}^{-10t}\left(30\sin\left(10t\right)-10\cos\left(10t\right)\right)$

$\text{Below is the graph of x vs t, }$

2 1:5 1- 0.5- 0.5

$\text{Below is the graph of v vs t, }$

0.5 -5 10
$\text{c)}$

$\text{For critical damping }\beta=4\sqrt{2}$

$\text{when }\beta=0\text{ equation will be }\frac{1}{5}x''+40x=0$

roots of auxilliary equation would be r.2-v8i

$\text{therefore, for resonance to happen the forcing function can be the options in (D) and (F)}$