Answer
Because the angle between ${\bf{w}}$ and ${\bf{N}}$ is obtuse, the vector ${\bf{w}}$ cannot be the acceleration vector.
Work Step by Step
Suppose ${\bf{w}}$ is the acceleration vector of a particle moving along the circle as is shown in Figure 14. Let $v\left( t \right)$ be the speed of the particle. By Eq. (1), we have
${\bf{w}} = {w_{\bf{T}}}{\bf{T}} + {w_{\bf{N}}}{\bf{N}}$,
where ${w_{\bf{T}}} = v'\left( t \right)$ ${\ }$ and ${\ }$ ${w_{\bf{N}}} = \kappa \left( t \right)v{\left( t \right)^2}$
Evaluate
${\bf{w}}\cdot{\bf{N}} = \left( {{w_{\bf{T}}}{\bf{T}} + {w_{\bf{N}}}{\bf{N}}} \right)\cdot{\bf{N}}$
${\bf{w}}\cdot{\bf{N}} = {w_{\bf{N}}} = \kappa \left( t \right)v{\left( t \right)^2}$
Since the angle between ${\bf{w}}$ and ${\bf{N}}$ is obtuse, the left-hand side is negative value. However, $\kappa \left( t \right) = \frac{1}{R}$, where $R$ is the radius of the circle, is always positive. So, the right-hand side is a nonnegative value. Thus, it is a contradiction. Hence, we conclude that the vector ${\bf{w}}$ cannot be the acceleration vector.