"Rotating wheel" is meant to indcate that the wheel is already rotating at the start. Denote the initial angular velocity by <em>ω₀</em>. Then the angular displacement <em>θ</em> at time <em>t</em> is
<em>θ</em> = <em>ω₀t</em> + 1/2 <em>αt</em> ²
while the angular velocity <em>ω</em> is
<em>ω</em> = <em>ω₀</em> + <em>αt</em>
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It takes 5 s for the wheel to rotate 38 times, or turn a total of (2<em>π</em> rad/rev) (38 rev) = 76<em>π</em> rad, as well as to reach an angular velocity of 79 rad/s, so that
76<em>π</em> rad = <em>ω₀</em> (5 s) + 1/2 <em>α</em> (5 s)²
79 rad/s = <em>ω₀</em> + <em>α</em> (5 s)
Solve the second equation for <em>ω₀</em> and substitute into the first equation, then solve for <em>α</em> :
<em>ω₀</em> = 79 rad/s - <em>α</em> (5 s)
==> 76<em>π</em> rad = (79 rad/s - <em>α</em> (5 s)) (5 s) + 1/2 <em>α</em> (5 s)²
==> 76<em>π</em> rad = (79 rad/s) (5 s) - 1/2 <em>α</em> (5 s)²
==> 1/2 <em>α</em> (5 s)² = (79 rad/s) (5 s) - 76<em>π</em> rad
==> <em>α</em> = ((79 rad/s) (5 s) - 76<em>π</em> rad) / (1/2 (5 s)²)
==> <em>α</em> ≈ 12.5 rad/s²
