A cylindrical log of wood of height H and area of cross-section A floa

1.	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 = \(2\pi\sqrt\frac{m}{Aρg}\)Where m is mass of the body and is density of the liquid.
Answer» 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. F_B = 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 = \(2\pi\sqrt{\frac{m}{Aρg}}\)

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 = \(2\pi\sqrt\frac{m}{Aρg}\)Where m is mass of the body and is density of the liquid.

Answer»

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.

F_B = 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 = \(2\pi\sqrt{\frac{m}{Aρg}}\)