4.16 Tangential and Normal Components of Acceleration
The acceleration vector \(\underline {a}\) can be resolved into its tangential and normal components as \[\underline {a} = \underline {a}_{\textbf {T}}\textbf {T} + \underline {a}_{\textbf {N}}\textbf {N}\hspace {0.1cm} , \hspace {0.2cm}\text {where}\] \[\underline {a}_{\textbf {T}} = \dfrac {d^2}{dt^2} = \frac {d}{dt}\left |\overline {V}\right |\hspace {0.2cm}\text {and}\] \[\underline {a}_{\textbf {N}} = K\hspace {0.1cm}\Bigg [\dfrac {ds}{dt}\Bigg ]^2 = K\left |\overline {V}\right |^2\] which are respectively, the tangential and normal components of acceleration. To calculate \(\underline {a}_{\textbf {N}}\), we use \[\underline {a}_{\textbf {N}} = \sqrt {\left |a\right |^2- \underline {a}_{\textbf {T}}^2 }\]
The position of a particle is \[\overline {r}(t) = \big (\cos t + t\sin t\big )\textbf {i} + \big (\sin t - t\cos t\big )\textbf {j}\] without finding \(\textbf {N}\) and \(\textbf {T}\), write \(\underline {a}\) in the form \(\underline {a} + \underline {a}_{\textbf {T}}\textbf {T} + \underline {a}_{\textbf {N}}\textbf {N}\).
\[\overline {V} = \big (t\cos t \big )\textbf {i} + \big ( t\sin t \big )\textbf {j}\]
\[\underline {a} = \big (\cos t - t\sin t\big )\textbf {i} + \big (\sin t + t\cos t \big ) \textbf {j}\]
\[\underline {a}_{\textbf {T}} = \dfrac {d}{dt}\left |\overline {V}\right | = \frac {d}{dt}\sqrt {t^2} = 1\]
\begin {align*} \left |\underline {a}\right |^ 2 & = \big (\cos t - t \sin t\big )^2 + \big ( \sin t + t\cos t\big )^2\\ & = \cos ^2 t - 2t\cos t \sin t + t^2\sin ^2 t + \sin ^2 t + 2t\sin t \cos t + t^2 \cos ^2 t\\ & = \big (\cos ^2t + \sin ^2t\big ) + t^2\big (\sin ^2t+\cos ^2 t\big )\\ & = 1 + t^2 \end {align*}
\begin {align*} \underline {a}_{\textbf {N}} & = \sqrt {\left |a\right |^2- \underline {a}_{\textbf {T}}^2 }\\ & = \sqrt {1 + t^2 - 1}\\ & = t \end {align*}
\[\therefore \hspace {0.5cm} \underline {a} = \underline {a}_{\textbf {T}}\textbf {T} + \underline {a}_{\textbf {N}}\textbf {N} =\textbf {T} + t\textbf {N}\]
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