A liquid is kept in a cylindrical vessel which is rotated along its axis. The liquid rises at the sides, if the radius of vessel is ` 0.05 m` and the speed of rotation is `2 rev//s`, find difference in the height of the liquid at the centre of the vessel and its sides.

Options

A. 0.001 m
B. 0.002 m
C. 0.01 m
D. 0.02 m

Answer

To solve the problem, we need to find the difference in height of the liquid in a cylindrical vessel that is rotating around its axis. Here’s a step-by-step solution:

### Step 1: Understand the parameters given
- Radius of the vessel, \( r = 0.05 \, \text{m} \)
- Speed of rotation, \( f = 2 \, \text{rev/s} \)

### Step 2: Convert the speed of rotation to angular velocity
Angular velocity \( \omega \) in radians per second can be calculated using the formula:
\[ \omega = 2 \pi f \]
Substituting the value of \( f \):
\[ \omega = 2 \pi \times 2 = 4 \pi \, \text{rad/s} \]

### Step 3: Calculate the linear velocity \( v \)
The linear velocity \( v \) at the edge of the vessel is given by:
\[ v = \omega r \]
Substituting the values of \( \omega \) and \( r \):
\[ v = 4 \pi \times 0.05 = 0.2 \pi \, \text{m/s} \]

### Step 4: Use the formula for height difference
The difference in height \( h \) of the liquid at the center and at the sides of the vessel is given by:
\[ h = \frac{v^2}{2g} \]
Where \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)).

### Step 5: Substitute \( v \) into the height difference formula
First, calculate \( v^2 \):
\[ v^2 = (0.2 \pi)^2 = 0.04 \pi^2 \, \text{m}^2/\text{s}^2 \]
Now, substitute \( v^2 \) into the height difference formula:
\[ h = \frac{0.04 \pi^2}{2 \times 10} = \frac{0.04 \pi^2}{20} = 0.002 \pi^2 \, \text{m} \]

### Step 6: Calculate the numerical value of \( h \)
Using \( \pi \approx 3.14 \):
\[ h \approx 0.002 \times (3.14)^2 = 0.002 \times 9.8596 \approx 0.0197 \, \text{m} \]
This can be approximated to:
\[ h \approx 0.02 \, \text{m} \]

### Conclusion
The difference in the height of the liquid at the center of the vessel and its sides is approximately \( 0.02 \, \text{m} \).

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