Two solid spheres, A and B , made of the same material, have radii r and 2r respectively. They are dropped into a large column of a viscous liquid. When they reach their respective terminal velocities, the rate of heat dissipation due to viscous drag for sphere A is P , and for sphere B is N P . The value of N is _____ .

Step-by-step solution

The rate of heat dissipation is equal to the mechanical power of the viscous force, which is given by P = F_ v v_ t . At terminal velocity, the viscous force balances the effective weight of the sphere (weight minus buoyant force): F_ v = W - F_ B = 4 3 r³ ( _ s - _ L ) g Thus, the viscous force is proportional to the cube of the radius: F_ v r³ . From Stokes' law, the terminal velocity is given by: v_ t = 2 9 r² ( _ s - _ L ) g Thus, the terminal velocity is proportional to the square of the radius: v_ t r² . The