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Question
A cylindrical log of wood of height h and area of cross-section A floats in water. It is pressed and then released. Show that the log would execute S.H.M. with a time period. `T = 2πsqrt(m/(Apg))` where m is mass of the body and ρ is density of the liquid.
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Solution
Consider the diagram,
Let the log be pressed and let the vertical displacement at the equilibrium position be x0.
At equilibrium, mg = buoyant force = (ρAx0)g ......[∵ m = Vρ = (Ax0)ρ]
When it is displaced by a further displacement x, the buoyant force is A(x0 + x)ρg
∴ Net restoring force = Buoyant force – Weight
= A(x0 + x)ρg – mg
= (Aρg)x .....[∵ mg = ρAx0g]
As displacement x is downward and restoring force is upward, We can write
`F_"restoring" = - (Aρg)x`
= `- kx`
Where k = constant = Aρg
So, the motion is S.H.M .....(∵ F ∝ – x)
Now, Acceleration a = `F_"restoring"/m = - k/m x`
Comparing with a = `- ω^2x`
⇒ `ω^2 = k/m`
⇒ `ω = sqrt(k/m)`
⇒ `(2pi)/T = sqrt(k/m)`
⇒ `T = 2pi sqrt(m/k)`
⇒ `T = 2pi sqrt(m/(Aρg))`
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