A closed tube filled with water is rotating uniformly in a horizontal plane about the axis OO' as shown in the figure. The manometers A and B which are fixed on the tube at distances r1 and r2 , indicate pressures P1 and P2 respectively. The angular velocity ( ω ) of the tube is

Question

A closed tube filled with water is rotating uniformly in a horizontal plane about the axis OO' as shown in the figure. The manometers A and B which are fixed on the tube at distances r1 and r2 , indicate pressures P1 and P2 respectively. The angular velocity ( ω ) of the tube is <img class="img-fluid question-image" alt="Question" src="https://cdn.tardigrade.in/q/nta/p-wpv3xagx9wrnhpar.jpg" /> (A) ω =√(2 (P2 - P1)/ρ (r22 - r12)) (B) ω =√(2 (P2 + P1)/ρ (r22 - r12)) (C) ω =√(2 (P2 + P1)/ρ (r22 + r12)) (D) ω =√(2 (P2 - P1)/ρ (r22 + r12))

Tardigrade Admin · Accepted Answer

Consider the condition of equilibrium for the mass of water contained between cross section separated by x and

x + d x

from rotation axis.

d p = ρ ω^{2} x d x

;

\int_{p_{1}}^{p_{2}} d p = \int_{r_{1}}^{r_{2}} ρ ω^{2} x d x

p_{2} - p_{1} = ρ ω^{2} [\frac{x ^{2}}{2}]_{r_{1}}^{r_{2}} = ρ ω^{2} (\frac{r _{2}^{2} - r _{1}^{2}}{2})

ω = \frac{2 ( P _{2} - P _{1} )}{ρ ( r _{2}^{2} - r _{1}^{2} )}

$ω = \frac{2 ( P _{2} - P _{1} )}{ρ ( r _{2}^{2} - r _{1}^{2} )}$

$ω = \frac{2 ( P _{2} + P _{1} )}{ρ ( r _{2}^{2} - r _{1}^{2} )}$

$ω = \frac{2 ( P _{2} + P _{1} )}{ρ ( r _{2}^{2} + r _{1}^{2} )}$

$ω = \frac{2 ( P _{2} - P _{1} )}{ρ ( r _{2}^{2} + r _{1}^{2} )}$

Q. A closed tube filled with water is rotating uniformly in a horizontal plane about the axis OO′ as shown in the figure. The manometers A and B which are fixed on the tube at distances r1​ and r2​ , indicate pressures P1​ and P2​ respectively. The angular velocity ( ω ) of the tube is

ω=ρ(r22​−r12​)2(P2​−P1​)​​

ω=ρ(r22​−r12​)2(P2​+P1​)​​

ω=ρ(r22​+r12​)2(P2​+P1​)​​

ω=ρ(r22​+r12​)2(P2​−P1​)​​

$ω = \frac{2 ( P _{2} - P _{1} )}{ρ ( r _{2}^{2} - r _{1}^{2} )}$

$ω = \frac{2 ( P _{2} + P _{1} )}{ρ ( r _{2}^{2} - r _{1}^{2} )}$

$ω = \frac{2 ( P _{2} + P _{1} )}{ρ ( r _{2}^{2} + r _{1}^{2} )}$

$ω = \frac{2 ( P _{2} - P _{1} )}{ρ ( r _{2}^{2} + r _{1}^{2} )}$