Question

Consider a water tank as shown in the figure. It's cross-sectional area is 0.4 m². The tank has an opening B near the bottom whose cross-section area is 1 cm². A load of 24 kg is applied on the water at the top when the height of the water level is 40 cm above the bottom, the velocity of water coming out the opening B is v ms\(^{-1}\). The value of v, to the nearest integer, is ________. [Take value of g to be 10 ms\(^{-2}\)] 

Hint:

The extra pressure from the load adds to the hydrostatic pressure (\(\rho gh\)) to increase the efflux velocity.

Answer 1

Correct Answer: 3

Step 1: Use Bernoulli's equation between the top surface (1) and the opening (2). \[ P_1 + \rho g h + \frac{1}{2} \rho v_1^2 = P_2 + 0 + \frac{1}{2} \rho v_2^2 \] Since \(A_{tank} \gg A_{hole}\), \(v_1 \approx 0\).
Step 2: Define pressures. \(P_2 = P_{atm}\). \(P_1 = P_{atm} + \frac{F}{A} = P_{atm} + \frac{24 \times 10}{0.4} = P_{atm} + 600 \text{ Pa}\).
Step 3: Substitute into Bernoulli's: \[ (P_{atm} + 600) + 1000 \times 10 \times 0.4 = P_{atm} + \frac{1}{2} \times 1000 \times v^2 \] \[ 600 + 4000 = 500 v^2 \implies 4600 = 500 v^2 \] \[ v^2 = 9.2 \implies v \approx 3.03 \text{ ms}^{-1} \] Nearest integer is 3.