The maximum speed and acceleration of a particle executing simple harmonic motion are $10 \mathrm{cm} {\mathrm{s}}^{-1}$ and $50 \mathrm{cm} {\mathrm{s}}^{-2}.$When the speed is $8 \mathrm{cm} {\mathrm{s}}^{-1}$ then the displacement of the particle from mean position is $\pm ...... \mathrm{cm}$

A particle having mass $10 \mathrm{g}$ oscillates according to the equation $x=(2.0 \mathrm{cm})\sin \left[\left(100 {\mathrm{s}}^{-1}\right)t+\frac{\pi }{6}\right].$ Find

(a) the amplitude, the time period and the spring constant,

(b) the position, the velocity and the acceleration at $t=0$.

The equation of motion of a particle started at $t=0$ is given by $x=5\sin (20t+\pi /3),$ where $x$ is in centimetre and $t$ in second. When does the particle

(a) first come to rest,

(b) first have zero acceleration,

(c) first have maximum speed?

Consider a particle moving in simple harmonic motion according to the equation
$x=2.0\cos \left(50\mathrm{\pi }t+{\tan }^{-1}0.75\right)$
where $x$ is in centimetre and $t$ in second. The motion is started at $t=0$.

(a) When does the particle come to rest for the first time?

(b) When does the acceleration have its maximum magnitude for the first time?

(c) When does the particle come to rest for the second time?