Question

Find \(\mathbf{T}(t), \mathbf{N}(t), a_{\mathrm{T}},\) and \(a_{\mathrm{N}}\) at the given time \(t\) for the plane curve \(\mathbf{r}(t)\) $$ \mathbf{r}(t)=\langle\omega t-\sin \omega t, 1-\cos \omega t\rangle, \quad t=t_{0} $$

Short answer

\(T(t)=\langle1 - cos(\omega t), sin(\omega t)\rangle\), \(N(t)=\langle sin(\omega t), cos(\omega t)\rangle\), \(a_{T}=0\), \(a_{N}=\omega\)

Step-by-step solution

Find \(r'(t)\)

Calculate the derivative of the vector function \(r(t)\), resulting in \(r'(t)= \langle \omega - \omega cos(\omega t), \omega sin(\omega t) \rangle\).

Find the magnitude of \(r'(t)\)

Calculate the magnitude of \(r'(t)\), hence getting \(||r'(t)|| = \sqrt{(\omega - \omega cos(\omega t))^2 + (\omega sin(\omega t))^2} = \omega\).

Find the unit tangent vector \(T(t)\)

Calculate the unit tangent vector \(T(t)\) which is given by \(T(t)=\frac{r'(t)}{||r'(t)||}\), resulting in \(T(t)=\langle1 - cos(\omega t), sin(\omega t)\rangle\).

Calculate derivative of \(T(t)\) and its magnitude

Calculate the derivative of the unit tangent vector \(T(t)\) and the magnitude of the derivative, hence getting \(T'(t)=\langle\omega sin(\omega t), \omega cos(\omega t)\rangle\) and \(||T'(t)|| = \sqrt{(\omega sin(\omega t))^2 + (\omega cos(\omega t))^2} = \omega\).

Find the normal vector \(N(t)\)

Calculate the normal vector \(N(t)\), given by \(N(t)=\frac{T'(t)}{||T'(t)||}\), resulting in \(N(t)=\langle sin(\omega t), cos(\omega t)\rangle\).

Calculate the tangential acceleration \(a_{T}\) and the normal acceleration \(a_{N}\)

Calculate the tangential acceleration \(a_{T}\), which is the derivative of the speed, \(||r'(t)||\), as a function of time, resulting in \(a_{T}=0\). Then calculate the normal acceleration \(a_{N}\), which is given by \(||T'(t)||\), hence getting \(a_{N}=\omega\).