The problem is as follows:
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.
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?.
let distance of point from centre of pulley be r let angular acc. = a write tangential acc. = r*a...........(1) write normal acc. = (w^2)*r.......(2) where w is angular velocity also w at t= 5 seconds is a*5 put w in (2) also tan(53)=normal acc./tangential acc. (this is because instantaneous linear velocity is along tangent at point) you will see that r is cancelled and a= 0.053