91Ó°ÊÓ

Given, spring force constant, k = 300 N/m

At time, t = 0s, the initial velocity of the block is $v_{0x}=-4m{s}^{-1}$,and the initial amplitude is $x_0=0.2m$.

Mass of the block, m = 2 kg

Formula of amplitude for SHM is,

$A=\sqrt{X_0^2+\frac{V_{0X}^2}{}}\left(1\right)$

Where, $x_0$is the displacement of the block, initial velocity of the block $v_{0x}$, and $\mathrm{Ó\neg }$is the angular frequency.

Now, $\mathrm{Ó\neg }$ in terms of force constant or the restoring force (k) and mass (m) of the block;

$\mathrm{Ó\neg }=\sqrt{\frac{k}{m}}\left(2\right)$

Equate the equation (1) & (2) for the value of A,

$$\begin{gathered} A=\sqrt{X_{0+}^2\frac{M{V}_{0x}^2}{k}} \\ =\sqrt{{\left(0.2m\right)}^2+\frac{2kg\left(-4m{s}^{-1}\right)}{300N{m}^{-1}}} \\ =0.383m \end{gathered}$$

Hence, the amplitude in SHM is $0.383m$.

The phase angle in SHM can be written as,

$\mathrm{Ï•}={\tan }^{-1}\left(-\frac{v_{0x}}{\mathrm{Ó\neg }{x}_0}\right)\left(3\right)$

Substitute the value of $\mathrm{Ó\neg }$ from equation (2) into (4), you get,

localid="1664184177082" $$\begin{gathered} \mathrm{Ï•}={\tan }^{-1}\left(-\frac{v_{0x}}{x_0}\sqrt{\frac{m}{k}}\right) \\ ={\tan }^{-1}\left(-\frac{-4m{s}^{-1}}{0.2m}\sqrt{\frac{2kg}{300N{m}^{-1}}}\right) \\ ={\tan }^{-1}\left(\frac{4m{s}^{-1}}{0.2m}\sqrt{\frac{2kg}{300N{m}^{-1}}}\right) \\ ={\tan }^{-1}\left(1.63rad\right) \\ =1.02rad \end{gathered}$$

Hence, the phase angle,localid="1668080027903" $\mathrm{Ï•}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{02}\mathbf{}\mathit{r}\mathit{a}\mathit{d}$.

The displacement x as a function of time can be written as,

$x=A\cos \left(\mathrm{Ó\neg }t-\mathrm{Ï•}\right)\left(4\right)$

Put the value of equation (2) or in equation (4), you get,

$$\begin{gathered} x=A\cos \left(\sqrt{\frac{k}{m}}t-\mathrm{ϕ}\right) \\ =0.383m×\cos \left(\sqrt{\frac{300N{m}^{-1}}{2kg}}t-1.02rad\right) \\ =0.383m×\cos \left(12.25rad{s}^{-1}×t-1.02rad\right) \end{gathered}$$

Hence, the equation for the position as a function of time

localid="1664184766171" $\mathit{x}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{383}\mathbf{}\mathit{m}\mathbf{×}\mathit{s}\mathit{i}\mathit{n}\left(12.25rad{s}^{-1}×t-1.02rad\right)$