The velocity of a small ball of mass $M$ and density $d_1$ when dropped in a container filled with glycerin becomes constant after some time. If the density of glycerin is $d_2$,the viscous force acting on the ball is ($g$ = acceleration due to gravity).

$A$ small spherical ball of radius $0.1 \,mm$ and density $10^{4} \,kg \,m^{-3}$ falls freely under gravity through a distance $h$ before entering a tank of water. If after entering the water the velocity of the ball does not change and it continues to fall with the same constant velocity inside the water,then the value of $h$ will be $m$. (Given $g = 10 \,m \,s^{-2}$,viscosity of water $\eta = 1.0 \times 10^{-5} \,N \,s \,m^{-2}$,density of water $\rho_w = 10^3 \,kg \,m^{-3}$)

Two solid spheres $S_{1}$ and $S_{2}$ of same uniform density fall from rest under gravity in a viscous medium and after some time, reach terminal velocities $v_{1}$ and $v_{2}$ respectively. If the ratio of masses $\frac{m_{1}}{m_{2}}=8$, then $\frac{v_{1}}{v_{2}}$ will be equal to

When a body falls in air,the resistance of air depends to a great extent on the shape of the body. $3$ different shapes are given in the figure: $(1)$ Disc,$(2)$ Ball,and $(3)$ Cigar-shaped. Identify the combination of air resistances $(R_1, R_2, R_3)$ which truly represents the physical situation. (The cross-sectional areas are the same).

If the terminal speed of a sphere $A$ [density $\rho_A = 7.5 \ kg \ m^{-3}$] is $0.4 \ ms^{-1}$ in a viscous liquid [density $\rho_L = 1.5 \ kg \ m^{-3}$],the terminal speed of sphere $B$ [density $\rho_B = 3 \ kg \ m^{-3}$] of the same size in the same liquid is: (in $ms^{-1}$)

In Millikan's oil drop experiment, a charged drop falls with terminal velocity $V$. If an electric field $E$ is applied in a vertically upward direction, then it starts moving in an upward direction with terminal velocity $2V$. If the magnitude of the electric field is decreased to $E/2$, then the terminal velocity will become