91Ó°ÊÓ

The one full swing takes 6seconds and the angle at the peak of her swing is $42°$ with the vertical. We have to build a model that relates her horizontal displacement after time twhen her swing is$5$feet long.

Let $t=0$, the maximum horizontal displacement of the swing from the rest position is the product of the length of the swing and sine angle made with the vertical.

So, $a=5\sin \left(42°\right)...........\left(i\right)$

Now, one full swing takes 6seconds so the period of the swing $T=6sec$

role="math" localid="1647605172267" $$\begin{gathered} T=\frac{2\mathrm{Ï€}}{\mathrm{Ó\neg }} \\ \mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{T} \\ \mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{6} \\ \mathrm{Ó\neg }=\frac{\mathrm{Ï€}}{3}...........\left(ii\right) \end{gathered}$$

The motion of the swing is harmonic motion. The equation of harmonic motion is given by $d=a\sin \left(\mathrm{Ó\neg }t\right)..........\left(iii\right)$

To model that relates her horizontal displacement after time tsubstitute the equation (i) and(ii)in (iii)

$d=5\sin \left(42°\right)×\sin \left(\mathrm{Ó\neg }t\right)$