A metal ball `B_(1)` (density `3.2g//"cc")` is dropped in water, while another metal ball `B_(2)` (density `6.0g//"cc")` is dropped in a liquid of density `1.6g//"cc"`. If both the balls have the same diameter and attain the same terminal velocity, the ratio of viscosity of water to that of the liquid is

Options

A. 2
B. 0.5
C. 4
D. indeterminate due to insufficient data

Answer

B

Explanation

The terminal velocity of the bodyof radius r, density `rho` falling through a medium of density `alpha` is given by
`v=2/9(r^(2)(rho-sigma_(water)g))/(eta)`
where `eta` is the coefficient of viscosity of the medium
`thereforevB_(1)=2/9(r_(B_(1))^(2))/(eta_("liquid"))(rho_(B_(2))sigma_("water"))g" "..(i)`
and `v_(B_(2))=(2)/(9)(r_(B_(1))^(2))/(eta_("liquid"))(rhoB_(2)-sigma_("liquid"))g" "...(ii)`
where the subscripts `B_(1)` and `B_(2)` respectively.
`because r_(B_(1))=r_(B_(2)) and v_(B_(1))=v_(B_(2))" "` (Given)
Substituting these values in (i) and (ii), we get
`(eta_("water"))/(eta_("liquid"))=((rho_(B_(1))-sigma_("water")))/((rho_(B_(2))-sigma_("liquid")))`
Substituting the given values, we get
`(eta_("water"))/(eta_("liquid"))=((3.2-1))/((6.0-1.6))(because` Density of water `=1 g cm^(-3))`
`(2.2)/(4.4)=0.5`