(A) The particle $P$ is performing uniform circular motion. At $t=0$,the position vector $OP$ makes an angle of $30^{\circ}$ (or $\pi/6$ radians) with

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$x(t)=r \cos \left(\omega t+\frac{\pi}{6}\right)$

(A) The particle $P$ is performing uniform circular motion. At $t=0$,the position vector $OP$ makes an angle of $30^{\circ}$ (or $\pi/6$ radians) with the positive $x$-axis.At any time $t$,the particle rotates by an angle $\omega t$ in the counter-clockwise direction.Therefore,the total angle $\theta$ made by the position vector $OP$ with the positive $x$-axis at time $t$ is $\theta = \omega t + 30^{\circ} = \omega t + \frac{\pi}{6}$.The projection of the position vector $OP$ on the $x$-axis is given by $x(t) = r \cos(\theta)$.Substituting the value of $\theta$,we get $x(t) = r \cos \left(\omega t + \frac{\pi}{6}\right)$.

Class 11 Physics Oscillations Periodic, Oscillatory motion and its characteristics and types of SHM and Equation of SHM

Easy

For particle $P$ revolving around the centre $O$ with radius of circular path $r$ and angular velocity $\omega$,as shown in the figure,the projection of $OP$ on the $x$-axis at time $t$ is .................

JEE MAIN 2023 All JEE MAIN

A
$x(t)=r \cos \left(\omega t+\frac{\pi}{6}\right)$
B
$x(t)=r \cos (\omega t)$
C
$x(t)=r \sin \left(\omega t+\frac{\pi}{6}\right)$
D
$x(t)=r \cos \left(\omega t-\frac{\pi}{6}\right)$

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Oscillations

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Periodic, Oscillatory motion and its characteristics and types of SHM and Equation of SHM

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$A$ particle free to move along the $x$-axis has potential energy given by $U(x) = k[1 - \exp(-x^2)]$ for $-\infty \le x \le +\infty$,where $k$ is a positive constant of appropriate dimensions. Then:

DifficultAIIMS 1995

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Draw a graph of displacement of $SHM$ as a function of time.

Medium

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Fill in the blanks:
$1.$ The ratio of displacement at any position and ....... remains constant for a particle executing $SHM$.
$2.$ The radius of the reference circle is equal to the .......... of the oscillator.
$3.$ Increase in phase per second of $SHO =$ ......... .
$4.$ $SHO$ covers ......... distance in one periodic time.

Medium

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Match the following functions with their corresponding nature of motion, where $\omega$ is a constant:

List-$I$ List-$II$

$A$. $\sin^2 \omega t$ $I$. Periodic but not $SHM$ $(T = 2\pi/\omega)$

$B$. $\sin^3 \omega t$ $II$. Periodic but not $SHM$ $(T = \pi/\omega)$

$C$. $\sin \omega t + \cos \pi \omega t$ $III$. Non-periodic

$D$. $\cos \omega t + \cos 2\omega t$ $IV$. Periodic but not $SHM$ $(T = 2\pi/\omega)$

DifficultJEE MAIN 2026

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The figure shows the circular motion of a particle. The radius of the circle,the period,the sense of revolution,and the initial position are indicated in the figure. The simple harmonic motion of the $x$-projection of the radius vector of the rotating particle $P$ is

Medium

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List-$I$	List-$II$
$A$. $\sin^2 \omega t$	$I$. Periodic but not $SHM$ $(T = 2\pi/\omega)$
$B$. $\sin^3 \omega t$	$II$. Periodic but not $SHM$ $(T = \pi/\omega)$
$C$. $\sin \omega t + \cos \pi \omega t$	$III$. Non-periodic
$D$. $\cos \omega t + \cos 2\omega t$	$IV$. Periodic but not $SHM$ $(T = 2\pi/\omega)$

For particle $P$ revolving around the centre $O$ with radius of circular path $r$ and angular velocity $\omega$,as shown in the figure,the projection of $OP$ on the $x$-axis at time $t$ is .................

Similar Questions

$A$ particle free to move along the $x$-axis has potential energy given by $U(x) = k[1 - \exp(-x^2)]$ for $-\infty \le x \le +\infty$,where $k$ is a positive constant of appropriate dimensions. Then:

Draw a graph of displacement of $SHM$ as a function of time.

The figure shows the circular motion of a particle. The radius of the circle,the period,the sense of revolution,and the initial position are indicated in the figure. The simple harmonic motion of the $x$-projection of the radius vector of the rotating particle $P$ is

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