A

Step 1: When the U-tube is rotated about an axis, the water experiences a centrifugal force due to the rotation, causing a change in the pressure distribution within the fluid.
Step 2: The pressure difference between the two limbs can be derived from considering the depth of water and the distance from the rotation axis. Let 'h' be the height of water in the stationary limb and 'd' be the distance from the rotation axis to the water level in the rotating limb.
Step 3: The pressure in the stationary limb can be expressed as:
$$ P_1 = ho g h $$
where 'P1' is the pressure at depth 'h', 'ρ' is the density of water, 'g' is acceleration due to gravity.
Step 4: For the rotating limb (let's call this height 'H') located at a distance 'R' from the rotation axis:
$$ P_2 = ho g H + rac{1}{2} ho rac{v^2}{R} $$
where 'v' is the tangential velocity, given by $v = Reta$, with 'β' being the angular velocity.
Step 5: Equating the two pressures, we get:
$$ ho g h = ho g H + rac{1}{2} ho rac{(R eta)^2}{R} $$
which simplifies to:
$$ g(h - H) = rac{1}{2} Reta^2 $$
Step 6: This equation allows us to find the relationship between the heights of water in the two limbs. The level of water in the limbs can be calculated knowing the angular velocity and the geometry of the system.
Conclusion: Therefore, the height difference due to the rotation can be derived using the principles of fluid statics and the effects of centrifugal motion.