$3\mathrm{.}1$ A guitar string vibrates harmonically with a frequency of $512$ Hz $($one octave above middle C on the musical scale$).$ If the amplitude of oscillation of the centerpoint of the string is $0\mathrm{.}002m(2mm),$ what are the maximum speed and the maximum acceleration at that point?

$3\mathrm{.}2$ A piston executes simple harmonic motion with an amplitude of $0\mathrm{.}1$ m $.$ If it passes through the center of its motion with a speed of $0\mathrm{.}5\frac{m}{s},$ what is the period of oscillation?

$3\mathrm{.}3$ A particle undergoes simple harmonic motion with a frequency of $10$ Hz $.$ Find the displacement $x$ at any time $t$ for the following initial condition:

$t=0,x=0\mathrm{.}25m,x^{}=0\mathrm{.}1\frac{m}{s}$