A cylindrical tank of height 1 m and cross section area A = 400 cm2 is initially empty when it is kept and a tap of cross sectional area 1 cm2. Water starts flowing from the tap at t = 0, with a speed = 2 m/s. There is a small hole in the base of the tank of cross-sectional area 0.5 cm2. The variation of height of water in the tank (in meters) with time t is best depicted by

Engineering

Physics

Liquid Pressure

Calculus

BernoulIis Equation and Equation of Continuity

Question

A cylindrical tank of height 1 m and cross section area A = 400 cm² is initially empty when it is kept and a tap of cross sectional area 1 cm². Water starts flowing from the tap at t = 0, with a speed = 2 m/s. There is a small hole in the base of the tank of cross-sectional area 0.5 cm². The variation of height of water in the tank (in meters) with time t is best depicted by

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The water level h increases due to inflow from tap (rate = a_inv_in) and decreases due to outflow from hole (rate = a_out√(2gh)). Initially, inflow dominates so h rises. As h increases, outflow velocity increases. Eventually, a steady state is reached when inflow equals outflow: a_inv_in = a_out√(2gh_max). The graph shows h increasing at a decreasing rate, approaching a constant maximum height asymptotically.

Steady state height: $h_{\max} = \frac{1}{2 g} {(\frac{a_{in} v_{in}}{a_{out}})}^{2}$

Final Answer: The second graph (top-right) correctly depicts the height approaching a steady maximum value asymptotically.