Given, m = mass of cylinder
h = height of cylinder
h1 = length of cylinder dipping in liquid in equilibrium position
ρ = density of liquid
A = area of cross section of cylinder
mg = buoyant force
= weight of water displaced by body
= ρ(Ah1)g …(i)
log is pressed gently through small distance x vertically and released.
FB = ρA(h1 + x)g
∴ Net restoring force, F = Buoyant Force – weight
= ρA(h1 + x)g − mg
= ρA(h1 + x)g − ρ(Ah1 )g [from (ii)]
= (Aρg)x
∴ F and x are in opposite direction.
F = −(Aρg)x
a = \(\frac{-(Aρg)}{m}x\) …(ii)
for standard SHM a = w2x …(iii)
∴ by (ii) & (iii) w2 = \(\frac{Aρg}{m}\)
or w = \(\sqrt{\frac{Aρg}{m}}\)
∴ T = \(2\pi\sqrt{\frac{m}{Aρg}}\)