phys問題題題2

Let the liquid level on each arm of the U-tube be displaced by an amount x , upward or downward, from the equilibrium position.

Pressure difference = (2x)pg, where g is the acceleration due to gravity
Restoring force = (2x)pgA, where a is the cross-sectional area of the U-tube
hence, [l.A.p]a = - (2x)pgA
where [l.A] is the volume of the liquid, and thus [l.A]p is the mass of liquid

a = - (2x)g/l = -(2g/l)x
i.e. the motion is simple harmonic with angular frequency square w^2 = 2g/l
Hence, period = 2.pi.square-root[l/2g]