A liquid is kept in a cylindrical vessel which is rotated along its axis. The liquid rises at the side. If the radius of the vessel is `5 cm` and the speed of rotation is `4 rev//s`, then the difference in the height of the liquid at the centre of the vessel and its sides is

Options

A. `8cm`
B. `2cm`
C. `40cm`
D. `4cm`

Explanation

To solve the problem of the height difference of the liquid in a rotating cylindrical vessel, we can follow these steps:

### Step 1: Understand the Problem
We have a cylindrical vessel with a radius of \( r = 5 \, \text{cm} = 0.05 \, \text{m} \) that is rotating at \( n = 4 \, \text{rev/s} \). We need to find the difference in height \( h \) between the center and the sides of the liquid in the vessel.

### Step 2: Convert Revolutions per Second to Angular Velocity
The angular velocity \( \omega \) in radians per second can be calculated using the formula:
\[ \omega = 2 \pi n \]
Substituting the given value:
\[ \omega = 2 \pi \times 4 = 8 \pi \, \text{rad/s} \]

### Step 3: Calculate the Linear Velocity
The linear velocity \( v \) at the radius \( r \) can be calculated using the formula:
\[ v = \omega r \]
Substituting the values:
\[ v = (8 \pi) \times (0.05) = 0.4 \pi \, \text{m/s} \]

### Step 4: Relate Velocity Head to Height Difference
The velocity head \( \frac{v^2}{2g} \) is converted into elevation head \( h \). Thus, we can write:
\[ h = \frac{v^2}{2g} \]
Substituting for \( v \):
\[ h = \frac{(0.4 \pi)^2}{2g} \]

### Step 5: Calculate \( g \)
Using \( g \approx 9.8 \, \text{m/s}^2 \):
\[ h = \frac{(0.4 \pi)^2}{2 \times 9.8} \]

### Step 6: Calculate \( h \)
Calculating \( (0.4 \pi)^2 \):
\[ (0.4 \pi)^2 = 0.16 \pi^2 \]
Now substituting back into the equation for \( h \):
\[ h = \frac{0.16 \pi^2}{19.6} \]
Using \( \pi^2 \approx 9.87 \):
\[ h \approx \frac{0.16 \times 9.87}{19.6} \approx \frac{1.5792}{19.6} \approx 0.0806 \, \text{m} \approx 8.06 \, \text{cm} \]

### Final Answer
Thus, the difference in height \( h \) between the center and the side of the liquid in the vessel is approximately:
\[ h \approx 8 \, \text{cm} \]

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