Question

Consider an object moving according to the position function \(\mathbf{r}(t)=a \cos \omega t \mathbf{i}+a \sin \omega t \mathbf{j}\) Determine the directions of \(\mathbf{T}\) and \(\mathbf{N}\) relative to the position function \(\mathbf{r} .\)

Short answer

The unit tangent vector \(\mathbf{T}\) points in the direction of motion, along the path of the object, and is perpendicular to the position vector \(\mathbf{r}\), while the unit normal vector \(\mathbf{N}\) always points toward the centre of the circle described by the position function \(\mathbf{r}\), and is perpendicular to both \(\mathbf{T}\) and \(\mathbf{r}\).

Step-by-step solution

Compute the velocity and acceleration vectors

First, find the derivative of the position function to get the velocity function. This involves differentiating each component separately using the chain rule. The derivative of \(\cos \omega t\) is \(- \omega \sin \omega t\) and the derivative of \(\sin \omega t\) is \(\omega \cos \omega t\). Then differentiate the velocity function to get the acceleration function. The derivative of \(-\omega \sin \omega t\) is \(-\omega^2 \cos \omega t\) and the derivative of \(\omega \cos \omega t\) is \(-\omega^2 \sin \omega t\).

Find the unit tangent and norm vectors

The unit tangent vector \(\mathbf{T}\) at any point can be obtained by normalizing the velocity vector \( \mathbf{v} \), that is \(\mathbf{T} = \frac{\mathbf{v}}{|\mathbf{v}|}\). The direction of the unit normal vector can be obtained by differentiating the unit tangent vector and normalizing the resulting vector.

Determine the direction relative to the position function

The position vector \(\mathbf{r}\) gives the location of a point in the plane. Compare the vectors \(\mathbf{r}\), \(\mathbf{T}\), and \(\mathbf{N}\). Determine how \(\mathbf{T}\) and \(\mathbf{N}\) are oriented with respect to \(\mathbf{r} .\) For example, if \(\mathbf{r}\) and \(\mathbf{T}\) are in the same direction, then \(\mathbf{T}\) points 'along' \(\mathbf{r}\). Or if \(\mathbf{r}\) and \(\mathbf{N}\) are orthogonal, then \(\mathbf{N}\) points 'toward the centre' of the path of the moving object.