(A) The equation of the parabola is $y = \frac{x^2}{5}$.For small oscillations,the slope is $\tan \theta = \frac{dy}{dx} = \frac{2x}{5}$.Since $\theta

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(A) The equation of the parabola is $y = \frac{x^2}{5}$.For small oscillations,the slope is $\tan \theta = \frac{dy}{dx} = \frac{2x}{5}$.Since $\theta$ is small,$\sin \theta \approx \tan \theta = \frac{2x}{5}$.The restoring force is $F = -mg \sin \theta = -mg \left( \frac{2x}{5} \right)$.Using Newton's second law,$m \frac{d^2x}{dt^2} = -mg \left( \frac{2x}{5} \right)$.$\frac{d^2x}{dt^2} = -\left( \frac{2g}{5} \right) x$.Comparing this with the $SHM$ equation $\frac{d^2x}{dt^2} = -\omega^2 x$,we get $\omega^2 = \frac{2g}{5}$.Given $g = 10 \ m/s^2$,$\omega^2 = \frac{2 \times 10}{5} = 4$.Therefore,$\omega = 2 \ rad/s$.

Class 11 Physics Oscillations Method to determine Time Period and Frequency for diffrent type of Object of SHM

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$A$ particle is placed at the lowest point of a smooth wire frame in the shape of a parabola,lying in the vertical $xy-$ plane having equation $x^2 = 5y$ ($x, y$ are in meters). After a slight displacement,the particle is set free. Find the angular frequency of oscillation in $rad/s$ (take $g = 10 \ m/s^2$).

A
$2$
B
$4$
C
$6$
D
$8$

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Method to determine Time Period and Frequency for diffrent type of Object of SHM

$A$ circular arc of mass $m$ is connected with the help of two massless strings as shown in the figure in a vertical plane. About point $P$,small oscillations are given in the plane of the arc. The time period of the oscillations of $SHM$ will be

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$A$ $U$-tube of a uniform bore has its arms vertical. The total length of the liquid in the two arms of the $U$-tube is $L$. The time period $T$ of the oscillation of the liquid column when it is displaced by $y$ is ($g$ = acceleration due to gravity).

DifficultMHT CET 2022

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$A$ particle executes linear $S.H.M.$ with amplitude $4 \ cm$. The magnitude of velocity and acceleration is equal when it is at $3 \ cm$ from the mean position. The time period of oscillation is:

MediumMHT CET 2025

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$A$ ring is hung on a nail. It can oscillate,without slipping or sliding $(i)$ in its plane with a time period $T_{1}$ and,$(ii)$ back and forth in a direction perpendicular to its plane,with a period $T_{2}$. The ratio $\frac{T_{1}}{T_{2}}$ will be

DifficultJEE MAIN 2020

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$A$ pendulum is formed by pivoting a disc. What distance from the center of mass should it be pivoted for the minimum time period while performing $SHM$?

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$A$ circular arc of mass $m$ is connected with the help of two massless strings as shown in the figure in a vertical plane. About point $P$,small oscillations are given in the plane of the arc. The time period of the oscillations of $SHM$ will be

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