One end of 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.

One end of U-tube containing mercury is connected to a suction pump an

1.	One end of 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.
Answer» Solution :The restoring force, F = - weight of mercury column at height 2y = -(VOLUME) `xx` DENSITY `xx` g = `-(Axx2yxxrhoxxg)=-2Arhogy` where A = area of cross-section of the tube, `RHO` = density of mercury. Force CONSTANT, `k=-F/y=2Arho*g` So, time period `T=2pisqrt(m/k)=2pisqrt(m/(2Arhog))` Let L = length of the whole mercury column. Therefore, mass of mercury, m = volume `xx` density = `ALrho` `therefore" "T=2pisqrt((ALrho)/(2Arhog))=2pisqrt(L/(2g))` where L is the total length of mercury column = 2h, h is the height of mercury column in U-tube. It shows that mercury column executes SHM.