(D) The frictional resistance $f_{r}$ is given as: $f_{r} \propto A$ and $f_{r} \propto v^{2}$ $f_{r} = k A v^{2}$ (where $k$ is a constant). The down

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(D) The frictional resistance $f_{r}$ is given as: $f_{r} \propto A$ and $f_{r} \propto v^{2}$ $f_{r} = k A v^{2}$ (where $k$ is a constant). The downward force (weight) is given as: $W = mg = \frac{4}{3} \pi r^{3} \rho g$. At terminal velocity,the downward force is balanced by the frictional resistance (ignoring buoyancy for simplicity as it is usually negligible in air): $mg = f_{r}$ $\frac{4}{3} \pi r^{3} \rho g = k (\pi r^{2}) v^{2}$ Simplifying the equation: $\frac{4}{3} r \rho g = k v^{2}$ Since $\rho$,$g$,and $k$ are constants,we have: $r \propto v^{2}$ $v \propto r^{1/2}$

Class 11 Physics Mechanical Properties of Fluids Viscosity and Stoke's Law and Terminal Velocity

Difficult

$A$ solid metallic sphere of radius $r$ is allowed to fall freely through air. If the frictional resistance due to air is proportional to the cross-sectional area and to the square of the velocity,then the terminal velocity of the sphere is proportional to which of the following?

A
$r^2$
B
$r$
C
$r^{3/2}$
D
$r^{1/2}$

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Mechanical Properties of Fluids

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Viscosity and Stoke's Law and Terminal Velocity

An air bubble of radius $1 \ cm$ rises from the bottom portion through a liquid of density $1.5 \ g/cc$ at a constant speed of $0.25 \ cm \ s^{-1}$. If the density of air is neglected, the coefficient of viscosity of the liquid is approximately, (in $Pa \ s$):

DifficultAP EAMCET 2006

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$A$ solid sphere falls with a terminal velocity of $10 \, cm/s$ in the Earth's gravitational field. If it is allowed to fall freely in a region outside the gravitational field of the Earth,the terminal velocity will be:

Easy

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$A$ steel ball of radius $0.05 \,cm$ and density $7.8 \,g \,cm^{-3}$ is dropped into a tank of water. The terminal velocity of the steel ball is (Density of water $= 1 \,g \,cm^{-3}$ and viscosity of water $= 0.001 \,Pa \,s$) (in $\,m/s$)

MediumAP EAMCET 2023

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$A$ metal sphere of radius $R$ and density $\rho_1$ is dropped in a liquid of density $\sigma$ and moves with terminal velocity $V$. Another metal sphere of the same radius and density $\rho_2$ is dropped in the same liquid. Its terminal velocity will be:

MediumMHT CET 2019

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$A$ small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density $\rho_0$. After some time,the ball falls with constant velocity. The viscous force on the ball is:

DifficultJEE MAIN 2024

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$A$ solid metallic sphere of radius $r$ is allowed to fall freely through air. If the frictional resistance due to air is proportional to the cross-sectional area and to the square of the velocity,then the terminal velocity of the sphere is proportional to which of the following?

Similar Questions

An air bubble of radius $1 \ cm$ rises from the bottom portion through a liquid of density $1.5 \ g/cc$ at a constant speed of $0.25 \ cm \ s^{-1}$. If the density of air is neglected, the coefficient of viscosity of the liquid is approximately, (in $Pa \ s$):

$A$ solid sphere falls with a terminal velocity of $10 \, cm/s$ in the Earth's gravitational field. If it is allowed to fall freely in a region outside the gravitational field of the Earth,the terminal velocity will be:

$A$ steel ball of radius $0.05 \,cm$ and density $7.8 \,g \,cm^{-3}$ is dropped into a tank of water. The terminal velocity of the steel ball is (Density of water $= 1 \,g \,cm^{-3}$ and viscosity of water $= 0.001 \,Pa \,s$) (in $\,m/s$)

$A$ metal sphere of radius $R$ and density $\rho_1$ is dropped in a liquid of density $\sigma$ and moves with terminal velocity $V$. Another metal sphere of the same radius and density $\rho_2$ is dropped in the same liquid. Its terminal velocity will be:

$A$ small ball of mass $m$ and density $\rho$ is dropped in a viscous liquid of density $\rho_0$. After some time,the ball falls with constant velocity. The viscous force on the ball is:

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