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

$1.52gm$

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B

$4.51gm$

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C

$3.15gm$

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D

$4.15gm$

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Solution

## The correct option is D $4.15gm$Step 1. Given dataRadius 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, ${F}_{b}=\frac{4}{3}\pi x{1}^{3}×1×9.8$Thus, $\begin{array}{l}m=\frac{\left(4/3\right)\pi {r}^{3}.{\rho }_{w}.g}{\left(g+a\right)}\\ =\frac{\left(4/3\right)\pi {\left(1\right)}^{3}.\left(1\right).980}{\left(980+9.8\right)}\\ =4.15gm\end{array}$Hence, option $\left(D\right)$ is correct.

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