The figure shows a U− tube of uniform cross-sectional area A having a liquid of mass M and density ρ. If the liquid in the column is slightly displaced, then it oscillates with a time period of : [g= acceleration due to gravity]
The correct option is DT=2π√M2Aρg If the level of liquid is depressed by y in column Q, then the level of liquid in column P will be 2y higher than the liquid level in column Q as shown in the figure.
Let m be the mass of increased height of liquid column in P. The weight of extra liquid in the column P, W=mg=ρVg=A(2y)ρg This will act as the restoring force on the entire liquid column. i.e Fr=−2Ayρg ∴ Acceleration (a)=−2AyρgM From a=−ω2y, ω=√2AρgM Thus, the liquid executes SHM. Time period of oscillation of liquid T=2πω=2π√M2Aρg Hence, option (d) is the correct answer.