Question

A cylindrical tank of radius 'R' and height 'H' is completely filled with a liquid of density '$\rho$'. A semicircular slit of vertical thickness $\delta$ (<<R) is cut open on the curved side wall, near the base of the tank, at time t = 0. Find the total reaction force on the tank due to the fan shaped water jet that comes out of the slit at t = 0. As shown in the figure, suppose the tank is free to slide on the frictionless fixed horizontal surface. Find the distance travelled by the tank till it becomes empty. Assume that its acceleration is very small throughout. The tank itself has negligible mass.

Answer 1

The reaction force is $\pi \rho g H \delta^2$. The distance travelled is indeterminate due to negligible mass and contradictory conditions.

Answer 2

The velocity of efflux from the slit is given by Torricelli's law: $v = \sqrt{2gH}$. The slit is a semicircle of vertical thickness $\delta$, implying its radius is $r = \delta$. The area of the semicircular slit is $A = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi \delta^2$. The reaction force on the tank is equal to the rate of change of momentum of the water jet. Assuming the water is ejected horizontally with velocity $v$, the force is $F = \rho A v^2$. Substituting the values: $F = \rho \left(\frac{1}{2} \pi \delta^2\right) (2gH) = \pi \rho g H \delta^2$.

For the second part, the tank has negligible mass. According to Newton's second law, $F = ma$. If $m \to 0$ and $F$ is finite and non-zero, the acceleration $a$ would approach infinity. This contradicts the condition that the acceleration is "very small throughout". This indicates an inconsistency in the problem statement for the second part, making the distance travelled indeterminate under the given conditions.