10.3 Worked Example - Angular Position from Angular Acceleration

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A diagram illustrating circular motion and acceleration.

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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