#39;. Tags and topics: JEE MAIN,JEE Advanced,JEE Mains,KCET,Mechanical Properties of Fluids,Viscosity,NCERT,NEET, JEE Main Physics,Class 11 Physics,Class 12 Physics.">
#39;. Tags and topics: JEE MAIN,JEE Advanced,JEE Mains,KCET,Mechanical Properties of Fluids,Viscosity,NCERT,NEET, JEE Main Physics,Class 11 Physics,Class 12 Physics.">
#39;. Tags and topics: JEE MAIN,JEE Advanced,JEE Mains,KCET,Mechanical Properties of Fluids,Viscosity,NCERT,NEET, JEE Main Physics,Class 11 Physics,Class 12 Physics.">
A small spherical ball of radius $r$, falling through a viscous medium of negligible density has terminal velocity '$v$'. Another ball of the same mass but of radius $2 r$, falling through the same viscous medium will have terminal velocity:
$4 \mathrm{v}$
$2 \mathrm{~V}$
$\frac{v}{4}$
$\frac{\mathrm{v}}{2}$
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Step 1: Identify the Relevant Principle
At low velocities, a sphere falling through a viscous medium experiences a drag force
governed by Stokes' law:
$F_{\text{drag}} = 6 \pi \eta r v,$
where $\eta$ is the viscosity of the fluid, $r$ is the radius of the sphere, and $v$ is the terminal velocity.
Step 2: Write the Force Balance at Terminal Velocity
At terminal velocity, the weight of the sphere (downward force) balances the viscous drag (upward force).
Since the fluid density is negligible, buoyant force is ignored. Therefore:
$mg = 6 \pi \eta r v,$
where $m$ is the mass of the sphere and $g$ is the acceleration due to gravity.
Step 3: Express the Terminal Velocity in Terms of the Radius
Solving for $v$, we get:
$v = \frac{mg}{6 \pi \eta r}.$
This shows $v$ is inversely proportional to the radius $r$ of the sphere.
In other words, $v \propto \frac{1}{r}.$
Step 4: Compare the Terminal Velocities
Let $v$ be the terminal velocity of the first sphere (radius $r$) and $v'$ be the terminal velocity of the second sphere (radius $2r$).
Using $v \propto \frac{1}{r},$ we have:
$\frac{v}{v'} = \frac{r'}{r} = \frac{2r}{r} = 2,$
which leads to
$v' = \frac{v}{2}.$
Step 5: State the Final Answer
Hence, the second ball with radius $2r$ attains a terminal velocity
$\frac{v}{2}.$