$3$

The beam AB is constructed from a lightweight material with Young's modulus $E.$ It has a

total length of $L$ and a second moment of area I.

$($a$)$ Consider Figure $9($a$),$ where a moment $M$ acts at point B$.$ Find the rotation of point B

in terms of $M,E,$I and $L.$

$[7$ marks$]$

$($b$)$ The rotational stiffness $k$ of the beam is defined as the ratio of moment $M$ to the

rotation of point B$.$ Express $k$ in terms of $E,$I and $L.$

$[1$ mark$]$

$($c$)$ Figure $9($b$)$ shows a heavy flywheel with mass $Q$ in the form of a solid circular disc of

radius $R,$ which is attached to point B using a bearing. While spinning at constant

angular velocity $,$ the bearing jams and the flywheel stops spinning. Find the resulting

rotation at point B in terms of $Q,R,$ and $k.$

$[4$ marks$]$

$($d$)$ Determine the vertical reaction force at A in terms of $Q,R,,E,$I and $L.$ Draw the

bending moment diagram of the beam.