A pulley slows down with an angular acceleration of -10pi rad/s2. If it stops after 20 s with a final angular speed of 20 rev/s, what was its initial angular speed and total angular displacement?

Hey I., good to see you again. Step 1: Identify the given values and convert units. The pulley slows down, so the angular acceleration is negative. • Final angular speed, _f = 20 rev/s • Angular acceleration, = -10 rad/s^2 (negative because it slows down) • Time, t = 20 s First, convert the final angular speed from revolutions per second to radians per second: _f = 20 revs × 2 rad1 rev = 40 rad/s Step 2: Calculate the initial angular speed (_i). Use the rotational kinematic equation: _f = _i + t Substitute the known values: 40 rad/s = _i + (-10 rad/s^2)(20 s) 40 rad/s = _i - 200 rad/s Solve for _i: _i = 40 rad/s + 200 rad/s _i = 240 rad/s The initial angular speed is 240 rad/s. Step 3: Calculate the total angular displacement (). Use the rotational kinematic equation: = _i t + (1)/(2) t^2 Substitute the known values: = (240 rad/s)(20 s) + (1)/(2)(-10 rad/s^2)(20 s)^2 = 4800 rad + (1)/(2)(-10 rad/s^2)(400 s^2) = 4800 rad - 2000 rad = 2800 rad Step 4: Convert the angular displacement to the number of revolutions. Since 1 revolution = 2 radians: Number of revolutions = ()/(2 rad/rev) Number of revolutions = 2800 rad2 rad/rev Number of revolutions = 1400 revolutions The number of revolutions made is 1400 revolutions. Send me the next one 📸