(A) In $SHM$,the displacement is given by $x = A \sin(\omega t)$. The velocity is $v = \frac{dx}{dt} = A\omega \cos(\omega t)$ and acceleration is $a

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(A) In $SHM$,the displacement is given by $x = A \sin(\omega t)$. The velocity is $v = \frac{dx}{dt} = A\omega \cos(\omega t)$ and acceleration is $a = \frac{dv}{dt} = -A\omega^2 \sin(\omega t)$.At the mean position,displacement $x = 0$,which implies acceleration $a = 0$ (minimum magnitude). At this point,velocity $v = A\omega$ (maximum magnitude). Thus,the Assertion is correct.Comparing $x = A \sin(\omega t)$ and $v = A\omega \sin(\omega t + \frac{\pi}{2})$,it is clear that the velocity leads the displacement by a phase difference of $\frac{\pi}{2}$. Thus,the Reason is also correct and explains why the velocity is maximum when displacement (and hence acceleration) is zero.

Class 11 Physics Oscillations Mix Examples-Oscillations

Medium

$Assertion :$ In simple harmonic motion,the velocity is maximum when the acceleration is minimum.
$Reason :$ Displacement and velocity of $S.H.M.$ differ in phase by $\frac{\pi }{2}$.

AIIMS 2014 All AIIMS

A
If both Assertion and Reason are correct and the Reason is a correct explanation of the Assertion.
B
If both Assertion and Reason are correct but Reason is not a correct explanation of the Assertion.
C
If the Assertion is correct but Reason is incorrect.
D
If both the Assertion and Reason are incorrect.

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Column $I$ gives a list of possible sets of parameters measured in some experiments. The variations of the parameters in the form of graphs are shown in Column $II$. Match the set of parameters given in Column $I$ with the graph given in Column $II$.

Column $I$ Column $II$

$(A)$ Potential energy of a simple pendulum ($y$-axis) as a function of displacement ($x$-axis) $(p)$ Parabolic curve opening upwards

$(B)$ Displacement ($y$-axis) as a function of time ($x$-axis) for a one-dimensional motion at zero or constant acceleration $(q)$ Linear graph passing through origin

$(C)$ Range of a projectile ($y$-axis) as a function of its velocity ($x$-axis) when projected at a fixed angle $(r)$ Linear graph with non-zero intercept

$(D)$ The square of the time period ($y$-axis) of a simple pendulum as a function of its length ($x$-axis) $(s)$ Parabolic curve opening upwards (starting from origin)

Column $I$	Column $II$
$(A)$ Potential energy of a simple pendulum ($y$-axis) as a function of displacement ($x$-axis)	$(p)$ Parabolic curve opening upwards
$(B)$ Displacement ($y$-axis) as a function of time ($x$-axis) for a one-dimensional motion at zero or constant acceleration	$(q)$ Linear graph passing through origin
$(C)$ Range of a projectile ($y$-axis) as a function of its velocity ($x$-axis) when projected at a fixed angle	$(r)$ Linear graph with non-zero intercept
$(D)$ The square of the time period ($y$-axis) of a simple pendulum as a function of its length ($x$-axis)	$(s)$ Parabolic curve opening upwards (starting from origin)

IIT 2008

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The displacement-time graph of a particle executing $S.H.M.$ is given in the figure: (sketch is schematic and not to scale). Which of the following statements is/are true for this motion?
$(A)$ The force is zero at $t = \frac{3T}{4}$
$(B)$ The acceleration is maximum at $t = T$
$(C)$ The speed is maximum at $t = \frac{T}{4}$
$(D)$ The $P.E.$ is equal to $K.E.$ of the oscillation at $t = \frac{T}{2}$

DifficultJEE MAIN 2020

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The amplitude of a particle executing $SHM$ about $O$ is $10 \, cm.$ Then:

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State whether the following statements are True or False:
$1.$ If a spring is cut into two equal pieces,the spring constant of each piece decreases.
$2.$ As the displacement of a Simple Harmonic Oscillator $(SHO)$ increases,its acceleration decreases.
$3.$ $A$ system can oscillate with more than one natural frequency.
$4.$ The periodic time of Simple Harmonic Motion $(SHM)$ depends on amplitude,energy,or phase constant.

Medium

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For a particle executing $S.H.M.$,where $x$ is the displacement from the equilibrium position,$v$ is the velocity at any instant,and $a$ is the acceleration at any instant,then:

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$Assertion :$ In simple harmonic motion,the velocity is maximum when the acceleration is minimum.$Reason :$ Displacement and velocity of $S.H.M.$ differ in phase by $\frac{\pi }{2}$.

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The amplitude of a particle executing $SHM$ about $O$ is $10 \, cm.$ Then:

For a particle executing $S.H.M.$,where $x$ is the displacement from the equilibrium position,$v$ is the velocity at any instant,and $a$ is the acceleration at any instant,then:

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$Assertion :$ In simple harmonic motion,the velocity is maximum when the acceleration is minimum.
$Reason :$ Displacement and velocity of $S.H.M.$ differ in phase by $\frac{\pi }{2}$.