Falling vertically in glycerine, a metal sphere has a mass of 50 mg and a radius of 1 mm . The coefficient of viscosity for glycerine at room temperature is 8.0 poise and its density is 1260 kg/m ^3 . Determine the terminal velocity at which the sphere will travel downward without acceleration.

Step-by-step solution

Given: Mass of the sphere, m = 50 mg = 50 10⁻⁶ kg Radius of the sphere, r = 1 mm = 10⁻³ m Coefficient of viscosity, = 8.0 poise = 0.8 Pa s Density of glycerine, = 1260 kg/m ^3 Acceleration due to gravity, g = 9.8 m/s ^2 Volume of the sphere, V = 4 3 r^3 Buoyant force, F_B = V g = 4 3 r^3 g Weight of the sphere, W = mg At terminal velocity v_t , the net downward force is balanced by the viscous drag force (Stokes' law): 6 r v_t = W - F_B v_t = mg - 4 3 r^3 g 6 r Calculating the mass of displaced glycerine: m' = 4 3