Question

A large water tank is fixed on a cart with wheels and a vane. The cart is tied to a fixed support with a rope. Water exits through a 5 cm diameter hole as a 10 m/s jet which is deflected by the vane by \(60^\circ\). The velocity of the jet after deflection remains 10 m/s. Density of water is \(1000\ \text{kg/m}^3\). The tension in the rope is \(\underline{\hspace{1cm}}\) N (round off to one decimal place). 

Hint:

Jet deflection problems use momentum change in the direction of interest. Use \(F = \dot{m}(V_{i} - V_{f})\).

Answer 1

Correct Answer: 98

Mass flow rate:
\[ A = \frac{\pi D^2}{4} = \frac{\pi (0.05)^2}{4} = 1.963\times10^{-3}\text{ m}^2 \] \[ \dot{m} = \rho A V = 1000(1.963\times10^{-3})(10) = 19.63 \text{ kg/s} \] Initial jet velocity (horizontal):
\[ \vec{V}_i = (10,\,0)\ \text{m/s} \] Final jet velocity after deflection at \(60^\circ\):
\[ \vec{V}_f = (10\cos 60^\circ,\,10\sin 60^\circ) = (5,\ 8.66)\ \text{m/s} \] Horizontal force exerted by fluid on vane:
\[ F_x = \dot{m}(V_{i,x} - V_{f,x}) = 19.63(10 - 5) \] \[ F_x = 19.63 \times 5 = 98.16\ \text{N} \] This horizontal force must be balanced by rope tension. Hence,
\[ T \approx 98.2\ \text{N} \]