What is the period of oscillation of a liquid contained in a U - tube executing shm ?

Options

A. 1.0 s
B. 1.54 s
C. 1.50 s
D. 2.0 s

Answer

B

Explanation

The x total mass of the vibrating system m= mass of mercury + mass of water= 1 kg + 0.2 kg = 1.2 kg
If the system is disturbed by a small displacement, Restoring force =`-m(d^(2)x)/(dt^(2))`
Also, the difference of levels in the two arms will be equivalent to the pressure difference of height of mercury (i.e.,) the restoring force F will be
`F=2xx0Ag` where p=13.6 is the relative density of mercury and A is area of cross-section of the tube.
Hence , `-m(d^(2)x)/(dt^(2))=F=2xpAg Rightarrow (-md^(2)x)/(dt^(2))=(2pAg)x`
Hence the period of the S.H.M.will be
`T=(2pi)/(omega)=2pi[sqrt((m)/(2pAg))]`
Putting values , m=12kg, `p=13.6xx10^(3)`, `A=pi0.6xx10^(-2))^(2)m^(2)`,`g=10ms^(-2)`
`T=sqrt(((12))/(2(13.6xx10^(3))(314)(0.6xx10^(-2))^(2)(10)))=154s`