A cube of density ρc floats in a liquid of density ρ1 as shown in the figure. At rest, an...
A cube of density ρc floats in a liquid of density ρ1 as shown in the figure. At rest, an amount h of the cube’s height is submerged in liquid. If the cube is pushed down, it bobs up and down like a spring and oscillates about its equilibrium position. Show that the frequency of its oscillations is given by f = (2π)−1 
Step-by-Step Solution
THINK:
When the cube is pushed into the liquid, cube will be acted by buoyant force acting against it. This will set cube in an oscillatory motion in the liquid. The density of liquid is 
, the density of cube is
, submerged height of the cube 
.
SKETCH:
The given diagram is shown below.
RESEARCH:
The mass displayed by the block is 
Buoyant force equal to the force of gravity on the displaced liquid 
The net force acting on the cube can be expressed as
The density of the cube and the liquid can be related as
The spring constant of suspended spring can be calculated as
The angular frequency of the cube can be expressed as
The mass of the displaced water is equal to the mass of the cube, since the mass of the displaced water can be calculated as
. . . . . . (1)
The mass of the cube can be expressed as
. . . . . . (2)
By equating equation (1) and (2), we can express the density of the cube as
SIMPLYFY:
The frequency of the oscillatory motion of the cube can be expressed as
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