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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