Answer
$$\|\mathbf{a}(t)\|=8.$$
Work Step by Step
We have $$
\mathbf{a}(t)=a_{\mathbf{T}}(t) \mathbf{T}(t)+a_{\mathbf{N}}(t) \mathbf{N}(t)
$$
and moreover, $v(t)=4$. Then $v'(t)=0$. Also, the circle of raduis $2$ has the curvature $\kappa(t)=\frac{1}{2}$, and $a_N=\kappa(t) v(t)^2$.
Now, we have
$$
\mathbf{a}(t)=0+\frac{1}{2} 4^2 \mathbf{N}(t)=8\mathbf{N}(t)
$$
and hence $$\|\mathbf{a}(t)\|=8.$$
