$A$ spring with a spring constant $1200 \; N m^{-1}$ is mounted on a horizontal table as shown in the figure. $A$ mass of $3 \; kg$ is attached to the free end of the spring. The mass is then pulled sideways to a distance of $2.0 \; cm$ and released. Determine:
$(i)$ the frequency of oscillations,
$(ii)$ maximum acceleration of the mass,and
$(iii)$ the maximum speed of the mass.

(N/A) Given:
Spring constant,$k = 1200 \; N m^{-1}$
Mass,$m = 3 \; kg$
Amplitude (displacement),$A = 2.0 \; cm = 0.02 \; m$
$(i)$ Frequency of oscillation $(v)$:
The frequency is given by $v = \frac{1}{2 \pi} \sqrt{\frac{k}{m}}$
$v = \frac{1}{2 \times 3.14} \sqrt{\frac{1200}{3}} = \frac{1}{6.28} \sqrt{400} = \frac{20}{6.28} \approx 3.18 \; Hz$
$(ii)$ Maximum acceleration $(a_{max})$:
The maximum acceleration is given by $a_{max} = \omega^2 A$,where $\omega = \sqrt{\frac{k}{m}}$
$a_{max} = \frac{k}{m} A = \frac{1200}{3} \times 0.02 = 400 \times 0.02 = 8.0 \; m s^{-2}$
$(iii)$ Maximum speed $(v_{max})$:
The maximum speed is given by $v_{max} = A \omega$
$v_{max} = A \sqrt{\frac{k}{m}} = 0.02 \times \sqrt{\frac{1200}{3}} = 0.02 \times 20 = 0.4 \; m s^{-1}$