A large compound pendulum is to be used as a tuned mass damper to suppress the

oscillations of a structure with a natural frequency of $0\mathrm{.}85$ Hz $.$

the required length of the pendulum can be calculated using

the following equation:

$L_{\text{b}\text{o}\text{b}}=\frac{1}{2}(\frac{g}{{}^2}+\sqrt[\phantom{2}]{(\frac{g}{{}^2}{)}^2-4\frac{m_{\text{r}\text{o}\text{d}}l}{m_{\text{b}\text{o}\text{b}}}(\frac{l}{3}-\frac{g}{2{}^2})})$

The pendulum rod has a length of $1\mathrm{.}2$ m and a mass of $0\mathrm{.}5$ kg $,$ and there are four masses

available to be used as the bob:

$m_1=3kg$

$m_2=4kg$

$m_3=5kg$

$m_4=6kg$

a$.$ The engineers would like to minimise the mass of the bob. Which of the four masses

should they use for the bob? Justify your answer.

b$.$ For the mass you identified use the equation to calculate $L_{\text{b}\text{o}\text{b}},$ the distance from the pin

to the centre of mass of the bob. Give your answer to two significant figures

c$.$ If the mass of the rod is negligible compared to the mass of the bob, show that the

equation for $L_{\text{b}\text{o}\text{b}}$ is consistent with the equation for a simple pendulum:

$f=\frac{1}{2}\sqrt[\phantom{2}]{\frac{g}{l}}$