(A) Given equation: $y = 5(\sin 3\pi t + \sqrt{3} \cos 3\pi t) \ cm$.We can rewrite this in the form $y = A \sin(\omega t + \phi)$.Multiply and divide

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(A) Given equation: $y = 5(\sin 3\pi t + \sqrt{3} \cos 3\pi t) \ cm$.We can rewrite this in the form $y = A \sin(\omega t + \phi)$.Multiply and divide by $2$: $y = 5 \times 2 \left( \frac{1}{2} \sin 3\pi t + \frac{\sqrt{3}}{2} \cos 3\pi t \right)$.Using the identity $\sin(A+B) = \sin A \cos B + \cos A \sin B$,we set $\cos \phi = 1/2$ and $\sin \phi = \sqrt{3}/2$,which gives $\phi = \pi/3$.So,$y = 10 \sin(3\pi t + \pi/3) \ cm$.The amplitude $A = 10 \ cm$.The angular frequency $\omega = 3\pi \ rad/s$.The time period $T = \frac{2\pi}{\omega} = \frac{2\pi}{3\pi} = \frac{2}{3} \ s$.

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

Medium

$A$ simple harmonic motion is represented by $y = 5(\sin 3\pi t + \sqrt{3} \cos 3\pi t) \ cm$. The amplitude and time period of the motion are:

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A
$10 \ cm, \frac{2}{3} \ s$
B
$10 \ cm, \frac{3}{2} \ s$
C
$5 \ cm, \frac{3}{2} \ s$
D
$5 \ cm, \frac{2}{3} \ s$

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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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Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion?
$(a)\; a=0.7 x$
$(b)\; a=-200 x^{2}$
$(c)\; a=-10 x$
$(d)\; a=100 x^{3}$

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The equation of motion of a particle is given by $x = a \sin(50t + \frac{\pi}{3}) \text{ cm}$. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are . . . . . . .

DifficultJEE MAIN 2026

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Equations $y = 2A \cos^2 \omega t$ and $y = A (\sin \omega t + \sqrt{3} \cos \omega t)$ represent the motion of two particles.

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The motion of a particle varies with time according to the relation $y = a(\sin \omega \,t + \cos \omega \,t)$,then

Easy

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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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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 $SHM$ $(T = 2\pi/\omega)$

$A$ simple harmonic motion is represented by $y = 5(\sin 3\pi t + \sqrt{3} \cos 3\pi t) \ cm$. The amplitude and time period of the motion are:

Similar Questions

Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion?$(a)\; a=0.7 x$$(b)\; a=-200 x^{2}$$(c)\; a=-10 x$$(d)\; a=100 x^{3}$

The equation of motion of a particle is given by $x = a \sin(50t + \frac{\pi}{3}) \text{ cm}$. The particle will come to rest at time $t_1$ and it will have zero acceleration at time $t_2$. The $t_1$ and $t_2$ respectively are . . . . . . .

Equations $y = 2A \cos^2 \omega t$ and $y = A (\sin \omega t + \sqrt{3} \cos \omega t)$ represent the motion of two particles.

The motion of a particle varies with time according to the relation $y = a(\sin \omega \,t + \cos \omega \,t)$,then

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Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion?
$(a)\; a=0.7 x$
$(b)\; a=-200 x^{2}$
$(c)\; a=-10 x$
$(d)\; a=100 x^{3}$