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

Options

A. `282.8 cm//sec`
B. `432.6 cm//sec`
C. `632.6 cm//sec`
D. `832.6 cm//sec`

Answer

To solve the problem of finding the speed at which the velocity head of a stream of water is equal to 40 cm of mercury, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Concept of Velocity Head**:
The velocity head (h_v) can be expressed in terms of the speed (V) of the fluid. The formula for velocity head is given by:
\[ h_v = \frac{V^2}{2g} \]
where \( g \) is the acceleration due to gravity (approximately \( 980 \, \text{cm/s}^2 \)).

2. **Convert the Given Height of Mercury to Pressure**:
The height of a mercury column can be converted into pressure. The pressure equivalent of a column of mercury (h) is given by:
\[ P = \rho g h \]
where \( \rho \) is the density of mercury (approximately \( 13.6 \, \text{g/cm}^3 \)) and \( h \) is the height in cm. For 40 cm of mercury:
\[ P = 13.6 \times 980 \times 40 \, \text{dyn/cm}^2 \]

3. **Set the Velocity Head Equal to the Pressure**:
Since we want the velocity head to be equal to the pressure head of 40 cm of mercury, we set:
\[ \frac{V^2}{2g} = 40 \, \text{cm of Hg} \]

4. **Convert 40 cm of Hg to Dynes**:
To find the equivalent in terms of cm/s², we can convert 40 cm of Hg to pressure:
\[ P = 13.6 \times 980 \times 40 \, \text{dyn/cm}^2 = 53328 \, \text{dyn/cm}^2 \]

5. **Solve for V**:
Now, we can set the two expressions equal to each other:
\[ \frac{V^2}{2g} = \frac{53328}{13.6} \]
Rearranging gives:
\[ V^2 = 2g \cdot \frac{53328}{13.6} \]
Substituting \( g = 980 \, \text{cm/s}^2 \):
\[ V^2 = 2 \times 980 \times \frac{53328}{13.6} \]
Calculate \( V \):
\[ V = \sqrt{2 \times 980 \times \frac{53328}{13.6}} \]

6. **Final Calculation**:
Performing the calculation gives the speed \( V \).