# Finding stopping time when only given initial angular velocity and an expression for angular acceleration?

Question:

A wheel starts is spinning at $27\text{ rad/s}$ but is slowing with an angular acceleration that has a magnitude given by $\alpha(t) = (3.0\;\mathrm{rad/s^4})t^2$. It stops in a time of: $\qquad$ ?

I happened to guess $3.0\text{ s}$ and it turned out to be right. But I want to understand why.

I tried using a kinematics equation and plugging $a$ in for $\alpha$, but it gave me the wrong answer. Am I missing something simple here?

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I figured it out. I needed to take the integral of the acceleration function, then since at t=0 w=27, my velocity function would be w = t^3 + 27. From there, it's just a simple matter of plugging in 0 for the final velocity and solving for t.