$(5$ points$)$ You are given the equation describing the time history of displacement of an undamped,

single degree of freedom system as $x(t)=$Asin$({}_{n}t+).$ If the absolute value of the amplitude of its

acceleration $($only its amplitude$)$ is measured to be $64m\frac{m}{s^2},$ and the absolute maximum value of

the amplitude of displacement, $A=4mm.$ Determine the following:

i$.(2$ points$)$ The expression for the velocity $x^{}(t)$ and acceleration $x^{}(t)$

ii$.(2$ points$)$ The natural frequency ${}_n$

iii. $(1$ point$)$ The spring constant, $k($consider mass $m=8kg)$

Hint: recall that the acceleration is the second derivative of displacement. Derive the expression

for the acceleration from the displacement and proceed from there.