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Step-by-Step Solution
Step 1: Identify the Physical Situation
A cylindrical vessel of cross-sectional area $A$ is placed on a horizontal surface. A small hole of cross-sectional area $a$ ($a \ll A$) is made at the bottom side. Liquid emerges horizontally from the hole. We need to determine the minimum coefficient of friction $\mu$ required so that the vessel does not slide due to the reaction (impact) of the fluid flow.
Step 2: Determine the Velocity of Efflux
For an ideal fluid flowing out from a small hole at the bottom of a vessel of liquid height $h$, the velocity of efflux $v$ is given by Torricelli’s law:
$v = \sqrt{2 g h}.$
Step 3: Calculate the Horizontal Momentum Flux
The mass of liquid flowing out per unit time (mass flow rate) is
$ \rho \, a \, v, $
where $ \rho $ is the density of the liquid, $a$ is the cross-sectional area of the hole, and $v$ is the velocity of efflux.
Hence, the horizontal force $F$ on the vessel due to the outflow of liquid (equal and opposite to the momentum flux) is:
$ F = \rho \, a \, v \times v = \rho \, a \, v^2. $
Step 4: Write the No-Sliding Condition
If the vessel does not slide, the frictional force $f$ must be at least equal to $F$. The maximum frictional force is given by
$ f_{\max} = \mu \, N = \mu \, mg, $
where $m$ is the mass of the vessel plus the liquid system, and $g$ is the acceleration due to gravity.
However, since the vessel is "light" and the main mass comes from the liquid of density $\rho$ and cross-sectional area $A$ filled to height $h$, the mass of the liquid is
$ m = \rho \, A \, h. $
Therefore,
$ f_{\max} = \mu \, (\rho \, A \, h) \, g = \mu \, \rho \, A \, h \, g. $
The no-sliding condition requires
$ \mu \, \rho \, A \, h \, g \ge \rho \, a \, v^2. $
Step 5: Substitute the Velocity of Efflux and Simplify
Substitute $v = \sqrt{2gh}$:
$ \rho \, a \, (\sqrt{2gh})^2 = \rho \, a \, (2gh). $
Hence,
$ \mu \, \rho \, A \, h \, g \ge \rho \, a \, (2 g h). $
Cancel common terms $\rho$, $g$, and $h$ on both sides:
$ \mu \, A \ge 2 \, a. $
Therefore,
$ \mu \ge \frac{2a}{A}. $
Step 6: State the Final Answer
The minimum coefficient of friction needed to prevent the vessel from sliding is
$ \displaystyle \frac{2a}{A}, $
which matches Option (3).
