Answer to Question #93266 in Classical Mechanics for Ayo

Question #93266
Determine the equation of motion of an oscillation in a U tube
1
Expert's answer
2019-08-26T10:43:31-0400



Let us consider a liquid with density ρ\rho in the U-tube of cross section area AA. The total length of liquid is ll. Let the liquid is displaced from equilibrium on distance y.y. The restoring force


Fy=Δmg=2yAρgF_y=-\Delta m g=-2yA\rho g

Using Newton's second law


my¨=Fym\ddot y=F_y

we obtain

my¨=2yAρgm\ddot y=-2yA\rho g

lAρy¨=2yAρglA\rho\ddot y=-2yA\rho g

Finally, the equation of motion of an oscillation in a U-tube


y¨+2gly=0\ddot y+\frac{2g}{l}y=0


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