Question

A liquid of mass m is oscillating with time period $T$ in a U-tube of area of cross-section A. If the liquid is placed in another tube of $\frac{A}{4}$ cross section area, then the time period will be

• A. $T$
• B. $2T$
• C. $\frac{T}{2}$
  
• D. none of these

Answer 1

The correct option is B $2T$

Time period of the oscillating liquid in a U-tube is given by $T=2\pi \sqrt{\frac{h}{g}}$ where $h$ is the height of the equilibrium liquid column.
If the same amount of liquid is placed in a tube of cross sectio area $\frac{A}{4}$, then liquid will rise to a height of $4h$ to keep the volume constant i.e. $A\times h=\frac{A}{4}\times 4h$
Therefore, the time period of the oscillating liquid in the new U-tube will be given by
$T^{{}^{\prime }}=2\pi \sqrt{\frac{4h}{g}}=2\left(2\pi \sqrt{\frac{h}{g}}\right)=2T$