A 1.0-m roulette wheel reaches a maximum angular speed of 18 rad/s before it begins decelerating. After reaching this maximum angular speed, it turns through 35 revolutions before it stops. How long did it take the wheel to stop after reaching its maximum angular speed?
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First, we need to convert the number of revolutions to radians, since we're dealing with angular speed. There are 2π radians in one revolution, so 35 revolutions is 35 * 2π = 70π radians.
We can use the formula for the final angular displacement under constant angular deceleration, which is θ = ωi*t + 0.5*α*t^2. In this case, the final angular displacement is 70π radians, the initial angular speed ωi is 18 rad/s, and the final angular speed is 0 (since the wheel stops), so the formula simplifies to 70π = 18t + 0.5*α*t^2.
We also know that the final angular speed is given by ωf = ωi + α*t. Since the final angular speed is 0, this simplifies to 0 = 18 + α*t.
Solving this equation for α gives α = -18/t.
Substituting this into the first equation gives 70π = 18t - 0.5*(18/t)*t^2. Simplifying this gives 70π = 18t - 9t, or 70π = 9t.
Finally, solving for t gives t = 70π/9 ≈ 24.6 seconds. So it took approximately 24.6 seconds for the wheel to stop after reaching its maximum angular speed.
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