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# One end of a U- tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns.

Q. 14.19 One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

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Let the height of each mercury column be h.

The total length of mercury in both the columns = 2h.

Let the cross-sectional area of the mercury column be A.

Let the density of mercury be $\rho$

When either of the mercury columns dips by a distance x, the total difference between the two columns becomes 2x.

Weight of this difference is $2Ax\rho g$

This weight drives the rest of the entire column to the original mean position.

Let the acceleration of the column be a Since the force is restoring

$\\2hA\rho (-a)=2xA\rho g\\ a=-\frac{g}{h}x$

$\frac{\mathrm{d}^{2}x }{\mathrm{d} t^{2}}=-\frac{g}{h}x$ which is the equation of a body executing S.H.M

The time period of the oscillation would be

$T=2\pi \sqrt{\frac{h}{g}}$

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