A U-tube of small and uniform cross-section c | vedclass

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Question

(B) Let the cross-sectional area be $A$ and the density of water be $\rho$. The total length of the water column is $L = 4H$. The mass of the water is

Vedclass · Accepted Answer

$\sqrt{\frac{gH}{8}}$

Vedclass · Answer

(B) Let the cross-sectional area be $A$ and the density of water be $\rho$. The total length of the water column is $L = 4H$. The mass of the water is $m = \rho A L = 4\rho AH$.Initially,the left column is at height $H/2$ above the mean level and the right column is at $H/2$ below the mean level. The potential energy of the system relative to the mean level is $U_i = \rho A (H/2) g (H/4) + \rho A (H/2) g (-H/4) = 0$ (considering the center of mass of each column).More simply,we can consider the potential energy change of the displaced water. $A$ volume $V = A(H/2)$ of water moves from the left side to the right side. The center of mass of this volume drops by a distance $H/2$. The change in potential energy is $\Delta U = -(\rho A (H/2)) g (H/2) = -\rho A g H^2 / 4$.By the law of conservation of energy,the loss in potential energy equals the gain in kinetic energy:$|\Delta U| = \frac{1}{2} m v^2$$\rho A g H^2 / 4 = \frac{1}{2} (4 \rho A H) v^2$$\rho A g H^2 / 4 = 2 \rho A H v^2$$v^2 = \frac{gH}{8}$$v = \sqrt{\frac{gH}{8}}$

Column $I$	Column $II$
$A$. Stoke's law	$I$. Pressure and energy
$B$. Turbulence	$II$. Hydraulic lift
$C$. Bernoulli's Principle	$III$. Viscous drag
$D$. Pascal's law	$IV$. Reynold's number

Column-$I$	Column-$II$
$(A)$ Hydraulic lift	$(P)$ Archimedes' principle
$(B)$ Razor blade can be made to float on the surface of water	$(Q)$ Pascal's law
$(C)$ The dam of a reservoir is made thick at the bottom	$(R)$ Surface tension
$(D)$ Ship is floating on the ocean	$(S)$ Pressure

Similar Questions

Match the columns $I$ and $II$:

Column $I$ Column $II$

$A$. Stoke's law $I$. Pressure and energy

$B$. Turbulence $II$. Hydraulic lift

$C$. Bernoulli's Principle $III$. Viscous drag

$D$. Pascal's law $IV$. Reynold's number

The correct match is:

$A$ vessel containing a liquid has a constant acceleration of $19.6 \, m/s^2$ in the horizontal direction. The free surface of the water gets sloped with the horizontal at an angle of ..........

$Assertion :$ $A$ thin stainless steel needle can float on a still water surface.
$Reason :$ Any object floats when the buoyancy force balances the weight of the object.

An air bubble of volume $50\, cm^3$ is formed at the bottom of a lake $47.6\, m$ deep. What will be its volume when it reaches the surface? (Atmospheric pressure $= 70\, cm$ of $Hg$ and density of $Hg = 13.6\, g/cm^3$)

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