Question

When a large bubble rises from bottom of a water lake to its surface, then its radius doubles. If the atmospheric pressure is equal to the pressure of height H of a certain water column then the depth of lake will be ______.

पर्याय

• 2H

• H

• 7H

• 4H

Answer 1

When a large bubble rises from bottom of a water lake to its surface, then its radius doubles. If the atmospheric pressure is equal to the pressure of height H of a certain water column then the depth of lake will be 7H.

Explanation:

When a large bubble rises from bottom of water lake to its surface, then its radius becomes double.

i.e., r₂ = 2r₁

Since, volume V ∝ r³

`therefore "V"_2/"V"_1 = ("r"_2/"r"_1)^3`

`= ((2"r"_1)/"r"_1)^3` = 8

`=> "V"_2 = 8"V"_1`    ....(i)

From Boyle's law,

p₁V₁ = p₂V₂

Where, p₂ = atmospheric pressure.

p₁V₁ = p₂ · 8V₁   ...[from Eq. (i)]

`"p"_2 = "p"_1/8`

Where, p₂ = atmospheric pressure.

and p₁ = pressure at depth d.

`therefore "p"_1 = "p"_"atmospherlc" + rho "gd"`

p₁ = p₂ + ρ gd

From Eqs. (Ii) and (iii), we have

`"p"_2 = ("p"_2 + rho "gd")/8`

7p₂ + ρ gd

7 · ρ gd = ρ gd

7 H = d

⇒ d = 7H