Question

An air bubble of radius $1cm$ in water has an upward acceleration $9.8cm{s}^{-2}$. The density of water is $1gmc{m}^{-3}$ and water offers negligible drag force on the bubble. The mass of the bubble is ($g=980cm/s^2$).

• A. $1.52gm$
• B. $4.51gm$
• C. $3.15gm$
• D. $4.15gm$

Answer 1

The correct option is D

$4.15gm$

Step 1. Given data

Radius of the air bubble, $r=1cm$

Upward acceleration, $a=9.8m{s}^{-2}$

Density of the water, ${\rho }_w=1gmc{m}^{-3}$

Acceleration due to gravity, $g=9.8m{s}^{-2}$

Step 2. Finding the mass of air bubble, $m$

Total force experienced by the air bubble as per our diagram is

$F_b–mg=ma$ [$F_b$is buoyant force is the product of the volume of the air bubble, density of the water and the acceleration due to the gravity.]

where, $F_b=\frac{4}{3}\pi x{1}^3\times 1\times 9.8$

Thus,

$\begin{array}{l}m=\frac{(4/3)\pi r^3.{\rho }_w.g}{(g+a)}\\ =\frac{(4/3)\pi {\left(1\right)}^3.\left(1\right).980}{(980+9.8)}\\ =4.15gm\end{array}$

Hence, option $\left(D\right)$ is correct.