(i) A non-uniform horizontal bar is supported on two springs, one of which is to the left of the...

(i) A non-uniform horizontal bar is
supported on two springs, one of which is to the left of the centre of gravity
and at a distance l1 from it, while the second is to the right of the centre of
gravity and at a distance l2 from it. The spring constants are k1 and k2,
respectively. The mass of the bar is m, while the moment of inertia about the
centre of gravity is Icg. Consider a reference frame with coordinates x
pointing down and θ denoting clockwise rotation of the bar about the
centre of gravity and show that the governing equations of motion are

(b) There exists a point
E, at a distance e to the right of the origin, where an applied external force
causes only a pure downward translation and an applied external moment causes a
pure rotation,

Hence discuss the
consequence of this result, i.e. the extent of the coupling between the two
modes of oscillation depends on the distance of the point E, representing the
point where system is statically uncoupled, and the centre of gravity C where
the system is dynamically coupled.