Water flows through a vertical tube of variable cross section. The cross-sectional areas at A and B are 4 mm ^2 and 2 mm ^2 respectively. If 1 cc of water enters through A per second and A is vertically above B by 15 16 cm , find the speed of flow at A , the speed of flow at B , and the pressure difference P_A - P_B . Take g = 10 m s ⁻² .

Step-by-step solution

Using continuity, Q = 1 cc/s = 10⁻⁶ m ^3 /s . At A , A_A = 4 10⁻⁶ m ^2 , so v_A = Q A_A = 10⁻⁶ 4 10⁻⁶ = 0.25 m/s = 25 cm/s . At B , A_B = 2 10⁻⁶ m ^2 , so v_B = Q A_B = 10⁻⁶ 2 10⁻⁶ = 0.5 m/s = 50 cm/s . By Bernoulli's equation, P_A + 1 2 v_A^2 + g z_A = P_B + 1 2 v_B^2 + g z_B . Since A is above B by 15 16 cm = 0.009375 m , P_A - P_B = 1 2 (v_B^2 - v_A^2) + g(z_B - z_A) . Substituting = 1000 kg/m ^3 , g = 10 m/s ^2 , v_A = 0.25 m/s and v_B = 0.5 m/s gives P_A - P_B = 93.75 - 93.75 = 0 . Hence the answers are 25 cm/