A certain CD has a playing time of 76 minutes. When the music starts, the CD is rotating at an angular speed of 4.8 102 revolutions per minute (rpm). At the end of the music, the CD is rotating at 2.1 102 rpm. Find the magnitude of the average angular acceleration of the CD. Express your answer in rad/s2.
<angular acceleration>=(final w -initial w)/time
change w to rad/sec before starting, and time in seconds.
To find the magnitude of the average angular acceleration of the CD, we need to use the formula:
Average angular acceleration = (final angular velocity - initial angular velocity) / time
First, we convert the initial and final angular velocities from revolutions per minute (rpm) to radians per second (rad/s).
Since 1 revolution is equal to 2π radians, we can convert from rpm to rad/s by multiplying the angular velocity by 2π/60.
Initial angular velocity (ω₁) = 4.8 * 10² rpm * (2π rad / 1 min) * (1 min / 60 s) = 4.8 * 10² * 2π rad/s
Final angular velocity (ω₂) = 2.1 * 10² rpm * (2π rad / 1 min) * (1 min / 60 s) = 2.1 * 10² * 2π rad/s
Now, let's calculate the average angular acceleration:
Average angular acceleration = (ω₂ - ω₁) / t
Since the time (t) is given in minutes and we want the answer in rad/s², we need to convert it to seconds.
Playing time = 76 minutes * (60 seconds / 1 minute) = 76 * 60 seconds
Now we can calculate the average angular acceleration:
Average angular acceleration = (2.1 * 10² * 2π - 4.8 * 10² * 2π) / (76 * 60)
Simplifying:
Average angular acceleration = (2.1 - 4.8) * 10² * 2π / (76 * 60)
Average angular acceleration ≈ (-2.7) * 10² * 2π / 4560
Average angular acceleration ≈ (-2.7) * 10² * π / 2280
Average angular acceleration ≈ (-2.7) * π / 22.8
Therefore, the magnitude of the average angular acceleration of the CD is approximately equal to (-2.7) * π / 22.8 rad/s².