# 10.3 Worked Example - Angular Position from Angular Acceleration

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A particle is moving in a circle of radius $$\displaystyle r$$ with an angular acceleration given by $$\displaystyle \vec{\alpha }(t)=\dfrac {1}{r}(A-Bt)\hat{k}$$, or $$\displaystyle a_{\theta }=(A-Bt)\hat{\theta }$$ where $$\displaystyle A$$ and $$\displaystyle B$$ are positive constants. At time $$\displaystyle t_0=0$$, the particle is at an angle $$\displaystyle \theta (0)$$ measured with respect to the $$\displaystyle +x$$-axis, and the tangential component of its velocity is $$\displaystyle v_{\theta }(0)=v_0$$.

Find the particle's angular position $$\displaystyle \theta (t)$$ as a function of time to obtain the arc length travelled by the particle during the time $$\displaystyle t$$ and defined as $$\displaystyle s(t)=r(\theta (t)-\theta (0))$$. Express your answer in terms of $$\displaystyle r$$, $$\displaystyle A$$, $$\displaystyle B$$, $$\displaystyle t$$, and $$\displaystyle v_0$$

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