Miscellaneous Examples of SHM

MOTION OF A LIQUID IN U-SHAPED TUBE

$\text{T = 2}\pi \sqrt{\frac{\text{m}}{\text{A}\rho g\left(\sin {\theta }_1+\sin {\theta }_2\right)}}$

m : mass of the liquid in the tube

$\rho$ : density of the liquid

A : cross sectional area of the tube

If the tube is of U-shape and liquid is filled to a height h,

then $\theta$₁ = $\theta$₂ = 90° and m = hAd x 2

$\therefore T=2\pi \sqrt{\frac{h}{g}}$

Exmaple : An ideal liquid of density $\rho$ occupies length $\ell$ of an U – tube of uniform cross-section as shown in the figure(10.25). When the liquid surface is displaced through x the liquid column executes SHM. Find the period of oscillation.

Solution: When the liquid column is displaced through x on the right side, the weight of the unbalanced liquid provides the restoring force.

$F=-\left[\rho A\left(2x\right)\right]g$

where A is the area of cross-section Applying Newton's second law

$F=m\frac{d^{2}x}{d{t}^2}=\left(\rho A\ell \right)\frac{d^{2}x}{d{t}^2}$

$\therefore \left(\rho A\ell \right)\frac{d^{2}x}{d{t}^2}+\rho A\left(2x\right)g=0$

or $\frac{d^{2}x}{d{t}^2}+\left(\frac{2g}{\ell }\right)x=0$

Time period is $T=2\pi \sqrt{\frac{\ell }{2g}}$

SHM OF A PARTIALLY SUBMERGED FLOATING OBJECT

$T=2\pi \sqrt{\frac{m}{A\rho \rho }}$

If it is connected to a vertical spring of spring constant k

$T=2\pi \sqrt{\frac{m}{k+A\rho \rho }}$