Question 2.SP.29: A wheel accelerates uniformly from rest to a speed of 200 rp......
A wheel accelerates uniformly from rest to a speed of 200 rpm in \frac{1}{2} s. It then rotates at that speed for 2 s before decelerating to rest in \frac{1}{3} s. How many revolutions does it make during the entire time interval?
Step-by-Step
\begin{array}{l}\text{From t = 0 to t = 0.5} :\qquad \qquad \theta_{1}=\frac{1}{2}\left(\omega_{0}+\omega\right) t=\frac{1}{2}(0+200 / 60 \mathrm{rev} / \mathrm{s})(0.5 \mathrm{~s})=0.83 \mathrm{rev} \\ \\ \text{From t = 0.5 to t = 2.5 :} \qquad \qquad \theta_{2}=\omega t=(200 / 60 \mathrm{rev} / \mathrm{s})(2 \mathrm{~s})=6.67 \mathrm{rev} \\ \\ \text{From t = 2.5 to rest:}\quad \qquad \qquad \theta_{3}=\frac{1}{2}\left(\omega_{0}+\omega\right) t=\frac{1}{2}(200 / 60 \mathrm{rev} / \mathrm{s}+0)(0.333 \mathrm{~s})=0.56 \mathrm{rev}\end{array}
Total number of revolutions
\theta=\theta_{1}+\theta_{2}+\theta_{3}=\underline{8.06 \mathrm{rev}}
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