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, $$F=a2h$$ ρ $$g=Mg=a3$$σ gIf it is pushed down by y then the buoyant force   F'$$=a2(h+y)$$ρ gRestoring force is  Δ$$F=F$$'−$$F=a2(h+y)$$σ g−$$a2h$$ σ g                                  $$=a2y$$ ρ gRestoring acceleration  $$=$$Δ$$FM=$$−$$a2y$$ρ $$gM=$$−$$a2$$ρ $$ga3$$σyMotion is S.H.M. ⇒$$T=2$$$$π$$$$a3$$σ$$a2$$ρ $$g=2$$$$π$$aσρ g

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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