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(D) For a simple harmonic motion,the displacement is given by $x = A \sin \omega t$.The velocity is $v = \frac{dx}{dt} = A \omega \cos \omega t = \ome

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(D) For a simple harmonic motion,the displacement is given by $x = A \sin \omega t$.The velocity is $v = \frac{dx}{dt} = A \omega \cos \omega t = \omega \sqrt{A^2 - x^2}$.The acceleration is $a = \frac{dv}{dt} = -A \omega^2 \sin \omega t = -\omega^2 x$.From the acceleration equation,we have $x = -\frac{a}{\omega^2}$.Substituting this into the velocity equation:$v^2 = \omega^2 (A^2 - x^2) = \omega^2 (A^2 - (-\frac{a}{\omega^2})^2) = \omega^2 A^2 - \omega^2 (\frac{a^2}{\omega^4}) = \omega^2 A^2 - \frac{a^2}{\omega^2}$.This equation is of the form $v^2 = -\frac{1}{\omega^2} a^2 + \omega^2 A^2$,which represents a straight line with a negative slope ($y = mx + c$,where $y = v^2$,$x = a^2$,$m = -\frac{1}{\omega^2}$,and $c = \omega^2 A^2$).Therefore,the graph of $v^2$ versus $a^2$ is a straight line with a negative slope,which corresponds to Graph $D$.

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)

$A$ graph of the square of the velocity against the square of the acceleration of a given simple harmonic motion is

Similar Questions

$A$ particle is executing $SHM$ with an amplitude $A$ and angular frequency $\omega$. The ratio of the maximum acceleration to the maximum velocity of the particle is:

The displacement-time graph of a particle executing $SHM$ is shown. Which of the following statements is/are true?

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