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Missing factor of S/s in hydrodynamics question

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A cylindrical vessel of height H and base area S is filled with water. An orifice of surface area $s \ll S$ is opened in the bottom of the vessel. Neglecting the viscosity of water, determine how soon all the water will pour out of the vessel.

Attempt :

Let at any instant height of water be h.

Let the speed from the bottom be $v$.

$v^2 = 2gh$

Differentating,

$2v \dfrac{dv}{dt}= 2g\dfrac{dh}{dt}$

$\implies v\dfrac{dv}{dt}= g\dfrac{dh}{dt}$

Now, $\dfrac{dh}{dt}= -v$

Substituting then integrating we get,

$v_i = gt$

$v_i = \sqrt{2gH}$

$\implies t = \sqrt{\dfrac{2H}{g}} $

But answer is: $t = \dfrac{S}{s}\sqrt{\dfrac{2H}{g}}$

Why am I missing that extra $S/s$ factor?

Please let me know my error. If my error is $v_2 \ne \dfrac{-dh}{dt} $, then please explain why.

asked Jul 21, 2018 in Physics Problems by Reststack (422 points)
edited Jul 21, 2018 by sammy gerbil

1 Answer

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Best answer

The velocity of the water leaving through the hole is $v=\sqrt{2gh}$. Your mistake is that this is not equal to $-\frac{dh}{dt}=V$ which is the velocity of the water level in the vessel.

Using the Continuity Equation the volume flow rate is the same at the water level and the hole. So the speeds $v, V$ are related by $$SV=sv$$ So $$\sqrt{2gh}=v=\frac{S}{s} V=-\frac{S}{s}\frac{dh}{dt}$$ $$dt=-\frac{S}{s}\frac{1}{\sqrt{2g}}\frac{dh}{\sqrt{h}}$$ $$t=\frac{S}{s}\frac{2\sqrt{H}}{\sqrt{2g}}=\frac{S}{s}\sqrt{\frac{2H}{g}}$$

answered Jul 21, 2018 by sammy gerbil (27,556 points)
selected Jul 23, 2018 by Reststack
Why should it not be equal to the -dh/dt? Intuitively, it is -dh/dt. (the velocity of the water coming out)
Reason: -v represents the rate at which the height is decreasing  i.e. dh/dt
If water is flowing through a pipe which gets narrower, the speed of the water is slow when the pipe is wide and fast when the pipe is narrow. $dh/dt$ is the speed in the wide part (the cylinder), the speed of the water level, which is slow. This is different from the speed at the narrow part (the hole), which is faster.
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