An open rectangular tank of water sits on a light trolley whose wheels run on…
PHYSICS · Multi-part · Fermi University Physics · Mechanics & Dynamical Systems · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow
Fluids and Continuum Physics, Ch Euler's equation and Bernoulli's theorem
Problem
An open rectangular tank of water sits on a light trolley whose wheels run on horizontal frictionless rails. A small hole of area 3.0 cm2 is opened in the left-hand vertical wall, with its centre 0.60 m below the free surface, and a horizontal jet issues from it. The combined mass of trolley, tank and water is 40 kg at the instant considered. Water density 1000 kg m−3, g=9.8 m s−2; atmospheric pressure acts on the free surface, on the jet and on the whole outside of the apparatus, and air resistance is ignored. Consider the instant just after the hole is opened. Treat the flow inside the tank as quasi-steady, inviscid and incompressible along a streamline from the free surface to the hole, take the tank to be wide enough that the free-surface speed is negligible, take the jet to leave horizontally with cross-sectional area equal to that of the hole, and neglect the loss of mass over this instant.
((a)) Find the speed of the jet as it leaves the hole.
((b)) Suppose for a moment that the trolley is held still. Apply the steady momentum balance to a fixed control volume that encloses the tank and its water and is crossed only by the jet, with gauge pressures on its surface, to find the horizontal force needed to hold the apparatus. Hence find the magnitude of the horizontal thrust the water exerts on the apparatus, and state its direction relative to the jet. Take this thrust to act unchanged just after the trolley is released.
((c)) Find the horizontal acceleration of the trolley at this instant.
((d)) Treat the water, apart from the small flow towards the hole, as moving rigidly with the trolley at this acceleration, and neglect the effect of that flow on the pressure in the bulk. Find the angle the free surface of the water makes with the horizontal.