Question

Consider an object moving according to the position function \(\mathbf{r}(t)=a \cos \omega t \mathbf{i}+a \sin \omega t \mathbf{j}\) Find \(\mathbf{T}(t), \mathbf{N}(t), a_{\mathbf{T}},\) and \(a_{\mathbf{N}}\)

Short answer

\(\mathbf{T}(t) = \mathbf{v}(t)/||\mathbf{v}(t)||\), \(\mathbf{N}(t) = \mathbf{T'}(t)/||\mathbf{T'}(t)||\), \(a_{\mathbf{T}} = \mathbf{a}.\mathbf{T}\), and \(a_{\mathbf{N}} = \mathbf{a}.\mathbf{N}\)

Step-by-step solution

Find Velocity

Differentiate the given position vector function \(\mathbf{r}(t)=a \cos \omega t \mathbf{i}+a \sin \omega t \mathbf{j}\) with respect to time \(t\) to find the velocity \(\mathbf{v}(t)\).

Compute Tangential Vector

Now find the tangential vector \(\mathbf{T}(t)\) by normalizing the velocity vector, i.e., \(\mathbf{T}(t) = \mathbf{v}(t)/||\mathbf{v}(t)||\).

Find Acceleration

Next, compute the acceleration \(\mathbf{a}(t)\) by differentiating the velocity vector \(\mathbf{v}(t)\) with respect to time \(t\).

Compute Normal Vector

Find the normal vector \(\mathbf{N}(t)\) that is orthogonal to the tangent vector \(\mathbf{T}(t)\). This is done by taking the cross product of \(\mathbf{T}(t)\) and its derivative, and normalizing the result. i.e., \(\mathbf{N}(t) = \mathbf{T'}(t)/||\mathbf{T'}(t)||\).

Calculate Tangential and Normal Accelerations

Finally, calculate the tangential acceleration \(a_{\mathbf{T}}\) and normal acceleration \(a_{\mathbf{N}}\) by taking the dot product of the acceleration vector \(\mathbf{a}(t)\) with the unit tangent vector \(\mathbf{T}(t)\) and the unit normal vector \(\mathbf{N}(t)\) respectively. i.e., \(a_{\mathbf{T}} = \mathbf{a}.\mathbf{T}\) and \(a_{\mathbf{N}} = \mathbf{a}.\mathbf{N}\).