A wind turbine rotates at 18.5 rpm and has an angular acceleration of 0.0605 rad/s2. If the wind turbine takes 32.0 s to come to a complete stop, how many revolutions will this take? (Enter your answer as a number without units.)
Question
A wind turbine rotates at 18.5 rpm and has an angular acceleration of 0.0605 rad/s2. If the wind turbine takes 32.0 s to come to a complete stop, how many revolutions will this take? (Enter your answer as a number without units.)
Solution
To solve this problem, we need to use the equations of rotational motion.
Step 1: Convert the rotational speed from rpm to rad/s. 1 rpm = 2π rad/60s. So, 18.5 rpm = 18.5 * 2π / 60 rad/s = 1.94 rad/s.
Step 2: Use the equation of motion: ω = ω0 + αt, where ω is the final angular speed, ω0 is the initial angular speed, α is the angular acceleration, and t is the time. Since the turbine comes to a stop, ω = 0.
So, 0 = 1.94 rad/s - 0.0605 rad/s^2 * 32.0 s. Solving for α, we get α = 1.94 rad/s / (0.0605 rad/s^2 * 32.0 s) = 1.0 rad/s^2.
Step 3: Use the equation θ = ω0t + 0.5αt^2, where θ is the angular displacement.
So, θ = 1.94 rad/s * 32.0 s - 0.5 * 1.0 rad/s^2 * (32.0 s)^2 = 62.08 rad.
Step 4: Convert the angular displacement from rad to revolutions. 1 revolution = 2π rad. So, 62.08 rad = 62.08 / 2π = 9.88 revolutions.
So, the wind turbine will make approximately 10 revolutions before it comes to a complete stop.
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