$1$ a$.$ Describe TWO $(2)$ examples each of positive and negative impacts of

vibration, and explain the importance of the study of vibration.

$[5$ marks$]$

b$.$ A mass $m$ is attached at the end of a massless rigid bar $PQ$ as shown in

FIGURE Q$1$b$.$ Given that mass $m=0\mathrm{.}1kg,$ stiffness coefficient $k_1=k_2=$

$500\frac{N}{m},$ length $l_1=0\mathrm{.}1m,l_2=0\mathrm{.}2$ and $l_3=0\mathrm{.}3m.$ Assume that the

angular displacement $$ at pin $P$ is small for this vibratory system.

i$.$ Draw the free body diagram of this vibratory system.

$[4$ marks$]$

ii$.$ Determine the equivalent spring constant at $Q.$

iii. Formulate the equation of motion for mass $m.$

iv$.$ Calculate the natural frequency of this vibratory system.