Question

An air bubble rises from the bottom of a deep lake the radius of the air bubble near the surface is 'r'. Choose the appropriate radius of the air bubble.

a) r/2 at depth 30m

b) r/2 at depth 70m

c) r/3 at depth 140m

d) r/3 at depth 260m

• A. a,c
• B. a,d
• C. b,c
• D. b,d

Answer 1

The correct option is D b,d

At a depth of h water pressure will be
$P_h=P_a+\rho gh$ eq(1)
where
$\rho =1000kg/m^3$
$g=9.81m/s^2$
$h=\text{depth of water bubble}$
$P_a={10}^{5}Pa$
initially bubble is below water at h, so
$P_1=P_h$
finally it rises to surface of lake so
$P_2=P_a$
we assume that temperature is constant when it is rising from bottom to surface
by ideal gas equation
$P_1{V}_1=P_2{V}_2$
volume can be written as $V=\frac{4}{3}\pi r^3$
where r is radius.
$P_h\times \frac{4}{3}\pi r_h^3=P_a\times \frac{4}{3}\pi r_s^3$
$r_h={\left(\frac{P_a}{P_h}\right)}^{1/3}r$
$r_h=\frac{r}{{\left(\frac{P_h}{P_a}\right)}^{1/3}}$
by eq(1)
$r_h=\frac{r}{{\left(\frac{P_a+\rho gh}{P_a}\right)}^{1/3}}$
$r_h=\frac{r}{{(1+0.0981h)}^{1/3}}$ eq(2)
(a) at depth 30m
h = 30m
put h=30 in eq(2)
$r_h=\frac{r}{1.5}$
false
(b) at depth 70m
h = 70m
put h=70 in eq(2)
$r_h=\frac{r}{2}$
true
(c) at depth 140m
h = 140m
put h=140 in eq(2)
$r_h=\frac{r}{2.4}$
false
(d) at depth 260m
h = 260m
put h=260 in eq(2)
$r_h=\frac{r}{3}$
true

Answer is D.