The maximum velocity of a simple harmonic motion represented by $y = 3\sin \left( 100t + \frac{\pi}{6} \right)$ is given by

$A$ spring executes $S.H.M.$ with mass $10 \ kg$ attached to it. The force constant of the spring is $10 \ N/m$. If at any instant its velocity is $40 \ cm/s$,the displacement at that instant is (Amplitude of $S.H.M.$ $= 0.5 \ m$) (in $m$)

Show that for a particle executing $SHM$,velocity and displacement have a phase difference of $\frac{\pi}{2}$.

In case of a simple harmonic motion, if the velocity is plotted along the $X$-axis and the displacement (from the equilibrium position) is plotted along the $Y$-axis, the resultant curve is an ellipse with the ratio:
Major axis (along $X$) $= 20 \pi \times$ Minor axis (along $Y$).
What is the frequency of the simple harmonic motion?

$A$ particle performs simple harmonic motion with amplitude $A$. Its speed is trebled at the instant that it is at a distance $\frac{2A}{3}$ from the equilibrium position. The new amplitude of the motion is

$A$ particle executes a linear $S.H.M.$ In two of its positions,the velocities are $V_1$ and $V_2$,and the accelerations are $a_1$ and $a_2$ respectively $(0 < a_1 < a_2)$. The distance between the positions is