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Question

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]



A
T=2πMg
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B
T=2πMAgρ
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C
T=2πMAρg
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D
T=2πM2Aρg
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Solution

The correct option is D T=2πM2Aρg
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=2πM2Aρg
Hence, option (d) is the correct answer.

Physics

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