91Ó°ÊÓ

Fluid dynamics fundamentally studies the behavior of liquids and gases in motion. When dealing with fluids like liquids in a rotating vessel, the behavior changes due to the opposing forces and pressures in the situation. When a cylindrical vessel rotates around its axis, the liquid inside it experiences a centrifugal force. This force pushes the liquid outward towards the sides of the vessel. As a result, the liquid rises at the sides and lowers at the center. This outward movement is largely influenced by the rotational speed of the vessel, affecting how the fluid redistributes itself.

Some key concepts in understanding fluid dynamics include:

• Pressure distribution: In a rotating fluid, pressure changes with respect to the radial distance from the center due to centrifugal forces.
• Centrifugal force: Acts outwards from the axis of rotation, affecting how the liquid's height varies across the vessel.

By understanding fluid dynamics, we gain insight into predicting the behavior of rotating liquids and can calculate metrics such as the height difference between the liquid's center and sides.

Angular velocity is an essential concept in rotations, defined as the rate of change of angular position of a rotating body. It is measured in radians per second (rad/s). In our problem, the angular velocity of the rotating vessel is key to determining how the fluid redistributes vertically.To calculate angular velocity when given revolutions per second, we convert it using the relation:

• 1 revolution = 2Ï€ radians.
• Angular velocity, \[\omega = n \times 2\pi\text{ rad/s},\]where \(n\) is the number of revolutions per second.

In this problem, the vessel rotates at 4 revolutions per second, leading to an angular velocity of:\[\omega = 4 \times 2\pi = 8\pi \text{ rad/s}.\]

Understanding angular velocity allows us to apply the formula for calculating the height difference in the fluid, as it directly influences the centrifugal force acting upon the liquid.

The height difference calculation is pivotal in understanding how the liquid's surface changes under rotation. With a rotating cylindrical vessel, the key task is to find how the height variation occurs from the center to the sides of the vessel.The formula for height difference \(\Delta h\) in a rotating liquid is:\[\Delta h = \frac{\omega^2 r^2}{2g},\]where:

• \(\omega\): Angular velocity of the vessel.
• \(r\): Radius of the vessel.
• \(g\): Acceleration due to gravity (approximately \(9.81 \text{ m/s}^2\)).

Substituting the given values:

• \(\omega = 8\pi\text{ rad/s}\)
• \(r = 0.05 \text{ m}\) (since 5 cm is converted to meters)

We find that:\[\Delta h = \frac{(8\pi)^2 \times (0.05)^2}{2 \times 9.81} \approx 0.08 \text{ m} = 8 \text{ cm}.\]
This calculation helps determine how much the liquid rises at the sides compared to the center, giving us a recognizable height difference of \(8 \text{ cm}\), confirming the correct option from the exercise.