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A U-tube is filled with a single homogeneous liquid. The liquid is temporarily depressed in one side by a piston. The piston is removed and the level of liquid in each side oscillates. Show that the period of oscillation is πsrc=data:image/png;base64,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, where L is the total length of the liquid in the tube.

Radhika Batra , 9 Years ago
Grade 11
anser 1 Answers
Jitender Pal

Last Activity: 9 Years ago

In an oscillating system, the restoring force F is defined as,
F = kx
Here k is the force constant and x is the displacement of the particle from its mean position.
The restoring force F acting on the liquid is defined as,
F = 2ρAxg
Here ρ is the density of the liquid, A is the area of cross section, x is the height from equilibrium position and g is the acceleration of free fall.
If we compare the equation F = kx with F = 2ρAxg, the force constant k will be,
k = 2ρAg
Density (ρ) is equal to mass (m) per unit volume (V).
ρ = m/ V
So using equation ρ = m/ V, mass m of the fluid in the U-tube having area A and total length of the liquid L will be,
ρ = m/ V
m= ρ V
= ρ(AL) (Since V= AL)
To find out the period of oscillation T, substitute ρ(AL) for m and 2ρAg for k in the equation T = 2π √m/k,
T = 2π √m/k
= 2π √( ρ(AL) /(2ρAg)
= π√2L/g
From the above observation we conclude that, period of oscillation will be π√2L/g.

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