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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 12 s\frac{1}{2}  s. It then rotates at that speed for 2 s before decelerating to rest in 13 s\frac{1}{3}  s. How many revolutions does it make during the entire time interval?

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From t = 0 to t = 0.5:θ1=12(ω0+ω)t=12(0+200/60 rev/s)(0.5 s)=0.83 rev From t = 0.5 to t = 2.5 :θ2=ωt=(200/60 rev/s)(2 s)=6.67 rev From t = 2.5 to rest:θ3=12(ω0+ω)t=12(200/60 rev/s+0)(0.333 s)=0.56 rev\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

θ=θ1+θ2+θ3=8.06 rev‾\theta=\theta_{1}+\theta_{2}+\theta_{3}=\underline{8.06  \mathrm{rev}}

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