# JEE Main & Advanced Physics Simple Harmonic Motion Various Formulae of S.H.M.

Various Formulae of S.H.M.

Category : JEE Main & Advanced

(1) S.H.M. of a liquid in U tube : If a liquid of density r contained in a vertical U tube performs S.H.M. in its two limbs. Then time period $T=2\pi \sqrt{\frac{L}{2g}}$$=2\pi \sqrt{\frac{h}{g}}$

where $L=$ Total length of liquid column,

$h=$ Height of undisturbed liquid in each limb $(L-2h)$

(2) S.H.M. of a floating cylinder : If $l$ is the length of cylinder dipping in liquid then Time period  $T=2\pi \sqrt{\frac{l}{g}}$

(3) S.H.M. of a small ball rolling down in hemi-spherical bowl $T=2\pi \sqrt{\frac{R-r}{g}}$

$R=$ Radius of the bowl

$r=$Radius of the ball

(4) S.H.M. of a piston in a cylinder $T=2\pi \sqrt{\frac{Mh}{PA}}$

$M=$ mass of the piston

$A=$ area of cross section

$h=$ height of cylinder

$P=$ pressure in a cylinder

(5) S.H.M. of a body in a tunnel dug along any chord of earth $T=2\pi \sqrt{\frac{R}{g}}$= 84.6 minutes

(6) Torsional pendulum : In a torsional pendulum an object is suspended from a wire. If such a wire is twisted, due to elasticity it exert a restoring toque $\tau =C\theta$. In this case time period is given by

$T=2\pi \sqrt{\frac{I}{C}}$

where $l=$ Moment of inertia a disc

$C=$ Torsional constant of wire $=\frac{\pi \eta {{r}^{4}}}{2l}$

$\eta =$ Modulus of elasticity of wire and

$r=$Radius of wire

(7) Longitudinal oscillations of an elastic wire : Wire/string pulled a distance $\Delta l$ and left. It executes longitudinal oscillations. Restoring force $F=-\,AY\,\left( \frac{\Delta l}{l} \right)$ $Y=$Young's modulus

$A=$ Area of cross-section

Hence $T=2\pi \sqrt{\frac{m}{k}}=2\pi \sqrt{\frac{ml}{AY}}$

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