A long cylindrical vessel is half filled with a liquid. When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. If the radius of vessel is 5 cm and its rotational speed is 2 rotations per second, then the difference in the heights between the centre and the sides, in cm, will be:a)1.2b)0.1c)2.0d)0.4Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Answer

Answer

Given:
- Radius of the vessel, r = 5 cm
- Rotational speed, n = 2 rotations per second

To find:
The difference in heights between the center and the sides of the vessel.

Explanation:
When the vessel is rotated about its own vertical axis, the liquid rises up near the wall. This phenomenon is known as the 'centrifugal force' or 'centrifugal effect.'

Centrifugal Force:
The centrifugal force is the apparent force that acts outward on a body moving around a center, arising from the body's inertia. In the case of rotating a vessel, the centrifugal force pushes the liquid away from the center towards the sides of the vessel.

Centrifugal Force Calculation:
The centrifugal force can be calculated using the following formula:

F = m * r * ω^2

Where:
- F is the centrifugal force
- m is the mass of the liquid
- r is the distance from the center of rotation to the liquid
- ω is the angular velocity (in radians per second)

Height Difference Calculation:
The height difference between the center and the sides of the vessel can be calculated using the following formula:

Δh = F / (ρ * g * A)

Where:
- Δh is the height difference
- ρ is the density of the liquid
- g is the acceleration due to gravity
- A is the cross-sectional area of the vessel

Calculating the Centrifugal Force:
The mass of the liquid can be calculated using the formula:

m = V * ρ

Where:
- V is the volume of the liquid
- ρ is the density of the liquid

The volume of the liquid can be calculated using the formula for the volume of a cylinder:

V = π * r^2 * h

Where:
- π is a mathematical constant (approximately equal to 3.14159)
- r is the radius of the vessel
- h is the height of the liquid

Substituting the values, we get:

V = π * (5 cm)^2 * (h cm)

The mass of the liquid is given by:

m = V * ρ = π * (5 cm)^2 * (h cm) * ρ

The centrifugal force can now be calculated using the formula:

F = m * r * ω^2 = π * (5 cm)^2 * (h cm) * ρ * 5 cm * (2 rotations/s)^2

Calculating the Height Difference:
Substituting the values of F, ρ, g, and A in the height difference formula, we get:

Δh = F / (ρ * g * A) = π * (5 cm)^2 * (h cm) * ρ * 5 cm * (2 rotations/s)^2 / (ρ * g * A)

The density of the liquid and the acceleration due to gravity cancel out, and the cross-sectional area of the vessel can be calculated using the formula for the area of a circle:

A = π * r^2 = π * (5 cm)^2

Substituting the values, we get:

Δh = π * (5 cm)^2 * (h cm) * 5 cm *

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