A pulley starts spinning from rest a rotation with constant angular acceleration. After $5\,s$ a point in its periphery

has an instant acceleration which makes a $53^{\circ}$ angle with its linear speed. Find the modulus of the angular acceleration (in $\frac{rad}{s^{2}}$) of the pulley.

has an instant acceleration which makes a $53^{\circ}$ angle with its linear speed. Find the modulus of the angular acceleration (in $\frac{rad}{s^{2}}$) of the pulley.

The given alternatives in my book are as follows:

$\begin{array}{ll}

1.&0.5\,\frac{m}{s}\\

2.&0.53\,\frac{m}{s}\\

3.&0.053\,\frac{m}{s}\\

4.&0.106\,\frac{m}{s}\\

5.&1.06\,\frac{m}{s}\\

\end{array}$

For this particular problem. I'm lost as how should I use the given information of the instant acceleration and the linear speed. How should I put those vectors?. Which sort of equation should I use?.

The only equation which comes to my mind for the angular acceleration is how it is related to the tangential acceleration as:

$a_{t}=\alpha \times r$

But in this case there is no radius.

Thus I believe it has something to do with vectors but I can't really find exactly how to use that information. Can somebody help me here?.