(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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Oscillations

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

$A$ particle executes $SHM$ in a line $4 \, cm$ long. Its velocity when passing through the centre of the line is $12 \, cm/s$. The period will be ..... $s$. (in $.047$)

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$A$ bead of mass $m$ is attached to the mid-point of a taut,weightless string of length $l$ and placed on a frictionless horizontal table. Under a small transverse displacement $x$,as shown in the figure,if the tension in the string is $T$,then the frequency of oscillation is:

KVPY 2010

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$A$ body executes simple harmonic motion under the action of a force $F_1$ with a time period $(4/5) \ s$. If the force is changed to $F_2$,it executes $SHM$ with a time period $(3/5) \ s$. If both the forces $F_1$ and $F_2$ act simultaneously in the same direction on the body,its time period (in $s$) is:

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One end of a $U$-tube containing mercury is connected to a suction pump and the other end to the atmosphere. $A$ small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the $U$-tube executes simple harmonic motion.

Medium

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$A$ rectangular block of mass '$m$' and cross-sectional area '$A$' floats on a liquid of density '$\rho$'. It is given a small vertical displacement from equilibrium, it starts oscillating with frequency ($g$ = acceleration due to gravity).

DifficultMHT CET 2026

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$A$ particle executes $SHM$ in a line $4 \, cm$ long. Its velocity when passing through the centre of the line is $12 \, cm/s$. The period will be ..... $s$. (in $.047$)

$A$ bead of mass $m$ is attached to the mid-point of a taut,weightless string of length $l$ and placed on a frictionless horizontal table. Under a small transverse displacement $x$,as shown in the figure,if the tension in the string is $T$,then the frequency of oscillation is:

One end of a $U$-tube containing mercury is connected to a suction pump and the other end to the atmosphere. $A$ small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the $U$-tube executes simple harmonic motion.

$A$ rectangular block of mass '$m$' and cross-sectional area '$A$' floats on a liquid of density '$\rho$'. It is given a small vertical displacement from equilibrium, it starts oscillating with frequency ($g$ = acceleration due to gravity).

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