A brass cube of side a and density σ is floating in mercury of density ρ. If the cube is displaced a bit vertically, it executes S.H.M. Its time period will be

Question

A brass cube of side a and density σ is floating in mercury of density ρ. If the cube is displaced a bit vertically, it executes S.H.M. Its time period will be

Infinity Learn · Accepted Answer

As a is the side of cube σ is its density.Mass of cube is $$a2$$ σ its weight $$=$$ $$a3$$σgLet h be the height of cube immersed in liquid of density ρ in equilibrium then, F $$=$$ $$a2h$$ρg $$=$$ Mg $$=$$ $$a3$$σgIf it is pushed down by y then the buoyant forceF' $$=$$ $$a2(h+y)$$ρgRestoring force is ∆F $$=$$ F'$$-F$$ $$=$$ $$a2(h+y)$$σ$$g-a2h$$σg                           $$=$$ $$a2y$$ρgRestoring acceleration $$=$$ ∆FM $$=$$ $$-a2y$$ρgM $$=$$ $$-a2$$ρ$$ga2$$σyMotion is S.H.M.⇒T $$=2$$$$π$$$$a3$$σ$$a2$$ρg $$=$$ $$2$$$$π$$aσρg

A brass cube of side a and density $σ$ is floating in mercury of density $ρ$ . If the cube is displaced a bit vertically, it executes S.H.M. Its time period will be

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