Two solid spheres A and B , made of the same material, are falling through a long column of a viscous liquid. The ratio of the radius of sphere A to that of sphere B is 1:2 . When both spheres have attained their respective terminal velocities, the ratio of the rate of heat generation (viscous power dissipation) of sphere A to that of sphere B is

Step-by-step solution

The rate of heat generation (power dissipated) by viscous drag is given by P = F_v v_t . According to Stokes' law, the viscous drag force is F_v = 6 r v_t . Thus, P = (6 r v_t) v_t = 6 r v_t^2 . The terminal velocity of a sphere falling in a viscous fluid is given by v_t = 2r^2g( _s - _l) 9 . For spheres of the same material in the same liquid, v_t r^2 . Substituting the proportionality of v_t into the power equation: P r(r^2)^2 P r^5 Given the ratio of radii is r_A : r_B = 1:2 , the ratio of power dissipation is: