Class 11

Physics

Mechanics

Mechanical Properties of Fluids

Two balls A and B have radii in the ratio 1:2. What will be the ratio of their terminal velocities in a liquid?

Let the radius of the smaller ball $=r$

the radius of the bigger ball$=2r$

Density of the ball's material$=ρ_{b}$

Density of liquid$=ρ$

Terminal velocity of the balls is v1 and v2 for the smaller and bigger respectively.

Volume of the smaller ball $V=34 πr_{3}$

The mass of the ball will be $=ρ_{b}V$

Volume of the bigger ball $=34 π(2r)_{3}=8V$

The mass of the ball will be$=8ρ_{b}V$

Upthrust on the smaller ball$=ρV_{g}$

Viscous drag force on smaller ball $=6πηrv_{1}$

Weight of the smaller ball $=ρ_{b}V_{g}$

Now, since the balls are moving with terminal velocity, the total upward force will balance its weight

$ρV_{g}+6πηrv_{1}=ρ_{b}V_{g} −−−−1.Upthrustonthebiggerball$= 8 \rho V_g$Viscousdragforceonbiggerball$= 6\pi \eta 2rv_2$Weightofthebiggerball$= 8\rho _bV_g$Similarly,$8\rho V_g+6\pi\eta 2rv_2=8\rho _bV_g$−−−−2.Dividingeqn.1byeqn.2$\dfrac{ρVg+6πηrv1}{8ρVg+12πηrv2}=\dfrac{1}{8}$=>$8\rho V_g+12\pi \eta rv_2=8\rho V_g+48\pi \eta rv_1$=>$12\pi \eta rv_2=48\pi \eta rv_1$=>$v_2=4v_1$=>$\dfrac{v_1}{v_2}=\dfrac{1}{4}$Thus,theratiooftheirterminalvelocity$= 1:4$$