A U-tube is of non uniform cross-section. The area of cross-sections of two sides of tube are A and 2 A (see fig.). It contains non-viscous liquid of mass m. The liquid is displaced slightly and free to oscillate. Its time period of oscillations is

Question

A U-tube is of non uniform cross-section. The area of cross-sections of two sides of tube are A and 2 A (see fig.). It contains non-viscous liquid of mass m. The liquid is displaced slightly and free to oscillate. Its time period of oscillations is (A) T=2 π √(m/3 ρ g A) (B) T=2 π √(m/2 ρ g A) (C) T=2 π √(m/ρ g A) (D) None of these

Tardigrade Admin · Accepted Answer

Suppose the liquid in left side limb is displaced slightly by

y

,
the liquid in right limb will increase by

y /2

.
The restoring force

F = - P A

= - ρ g (\frac{3 y}{2}) \times 2 A = 3 ρ g A (- y) .

a = \frac{F}{m} = 3 ρ g A (- y) / m

On comparing with,

a = - ω^{2} y

, we get

ω = \frac{3 ρ g A}{m}

and

T = 2 π \frac{m}{3 ρ g A}

Q. A U-tube is of non uniform cross-section. The area of cross-sections of two sides of tube are $A$ and $2 A$ (see fig.). It contains non-viscous liquid of mass $m$ . The liquid is displaced slightly and free to oscillate. Its time period of oscillations is

$T = 2 π \frac{m}{3 ρ g A}$

$T = 2 π \frac{m}{2 ρ g A}$

$T = 2 π \frac{m}{ρ g A}$

None of these

Q. A U-tube is of non uniform cross-section. The area of cross-sections of two sides of tube are A and 2A (see fig.). It contains non-viscous liquid of mass m. The liquid is displaced slightly and free to oscillate. Its time period of oscillations is

T=2π3ρgAm​​

T=2π2ρgAm​​

T=2πρgAm​​

None of these

Suppose the liquid in left side limb is displaced slightly by y, the liquid in right limb will increase by y/2. The restoring force F=−PA =−ρg(23y​)×2A=3ρgA(−y). a=mF​=3ρgA(−y)/m On comparing with, a=−ω2y, we get ω=m3ρgA​​ and T=2π3ρgAm​​

Q. A U-tube is of non uniform cross-section. The area of cross-sections of two sides of tube are $A$ and $2 A$ (see fig.). It contains non-viscous liquid of mass $m$ . The liquid is displaced slightly and free to oscillate. Its time period of oscillations is

$T = 2 π \frac{m}{3 ρ g A}$

$T = 2 π \frac{m}{2 ρ g A}$

$T = 2 π \frac{m}{ρ g A}$