Question

A U-tube filled with liquid is being rotated about an axis with angular speed \( \omega \) as shown in the figure. The difference in heights of liquid in two arms is

(1) \( \frac{\omega^{2}}{2 g}\left(r_{1}^{2}-r_{2}^{2}\right) \)

(2) \( \frac{\omega^{2}}{g}\left(r_{1}^{2}-r_{2}^{2}\right) \) 

(3) \( \frac{2 \omega^{2}}{g}\left(r_{1}^{2}+r_{2}^{2}\right) \) 

(4) \( \frac{\omega^{2}}{2 g}\left(r_{1}^{2}+r_{2}^{2}\right) \)

Answer 1

Net radially inward thrust on the element 

= (P + dp) A - PA

dm w²r = (P+dp) - p

P₁ - P₂ = \(∫w^2∫^{r_1}_{-r_{2}}rdr\)

P₁ - P₂ = \(\frac{∫w^2}{2}(r^2_1-r^2_2)\)

h₁ - h₂ = \(\frac{w^2}{2g}(r^2_1-r^2_2)\)