A given shaped glass tube having uniform cross-section is filled with water and is mounted on a rotatable shaft as shown in figure. If the tube is rotated with a constant angular velocity `omega` then :

Options

A. water levels in both sections A and B go up
B. water level in section A goes up and that in B comes down
C. water level in section A comes down and that in B it goes up
D. water levels remain same in both sections

Answer

A

Explanation

Due to rotation about the axis OO', a force is acting on the luqiid column AB outwards, pushing up the liquid column on the right. This centrifugal force acting on the right. This centrifugal force acting on the small element dx f area of cross- section dx, distant x from the axis is `dm omega^(2)x=A dx rho omega^(2)x`
The total force due to column of length L(=AB) is
`overset(L)underset(0)A rho omega^(2) cdot xdx = A rho omega^(2)(L^(2))/(2)`
Pressure at B= atmospheric pressure `+h_(1)rhog+(A rho omega^(2))/(A)(L^(2))/(2)`
Pressure at B due to liquid column on the right hand side = atmospheric pressure `+h_(2)rho g`
Net pressure due to the left hand side = pressure at B due to liquid column
atm, pressure `+h_(1)rho g+rho omega^(2)(L^(2))/(2)=` atm. pressure `+h_(2)rho g`
`therefore" "h_(2)-h_(1)=H_(0)=(omega^(2)L^(2))/(2g)`
Here one has

`H_(2)=(omega^(2)t_(2)^(2))/(2g), H_(1)=(omega^(2)L_(1)^(2))/(2g)`
As `omega` is the same, and `L_(2) gt L_(1), H_(2)gt H_(1)`. But Both will go up.