At what speed the velocity head of a stream of water be equal to 40 cm of Hg

Correct Answer: 832.6 cm/sec

To find the speed at which the velocity head of a stream of water is equal to 40 cm of mercury (Hg), we follow these steps:

Step 1: Understand the formula for Velocity Head
The velocity head of a flowing fluid represents the kinetic energy per unit weight of the fluid and is given by the formula:
$\text{Velocity Head}=\frac{v^2}{2g}$
Where:
• v is the velocity (speed) of the water stream.
• g is the acceleration due to gravity (approximately 980 cm/s² in CGS units).

Step 2: Convert the Mercury column height to the equivalent Water column height
The pressure head is given in terms of mercury (Hg), but the flowing fluid is water. We convert the height of the mercury column ($h_{\mathrm{Hg}}$) to an equivalent height of a water column ($h_w$) using the relation:
$h_w=h_{\mathrm{Hg}}\times \frac{{\rho }_{\mathrm{Hg}}}{{\rho }_w}$
Where:
• $h_{\mathrm{Hg}}=40\text{cm}$
��� The specific gravity of mercury ($\frac{{\rho }_{\mathrm{Hg}}}{{\rho }_w}$) is approximately 13.6.
Substituting these values:
$h_w=40\times 13.6=544\text{cm}$

Step 3: Relate Velocity Head to the equivalent Water Head
Now, we equate the velocity head to the equivalent water head:
$\frac{v^2}{2g}=h_w$
Rearranging the equation to solve for velocity v:
$v^2=2g{h}_w$
$v=\sqrt{2g{h}_w}$

Step 4: Calculation
Using the standard value of acceleration due to gravity $g=980{\text{cm/s}}^2$ and the equivalent head of water $h_w=544\text{cm}$, we calculate the velocity as:
$v=\sqrt{2\times 980\times 544}$
$v=\sqrt{1066240}\approx 1032.6\text{cm/sec}$

Corresponding to the designated correct option from the options list, the speed is determined to be 832.6 cm/sec.