A certain CD has a playing time of 68 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.
To find the magnitude of the average angular acceleration of the CD, we can use the following formula:
Average angular acceleration = (final angular speed - initial angular speed) / time
Given:
Initial angular speed = 4.8 * 10^2 rpm
Final angular speed = 2.1 * 10^2 rpm
Time = 68 minutes
First, we need to convert the angular speeds from rpm to rad/s.
1 revolution = 2π radians
1 minute = 60 seconds
Initial angular speed in rad/s:
Initial angular speed = 4.8 * 10^2 rpm * 2π rad/revolution * 1 minute / 60 seconds
= 4.8 * 10^2 * 2π / 60 rad/s
= 160π rad/s (approximately 502.65 rad/s)
Final angular speed in rad/s:
Final angular speed = 2.1 * 10^2 rpm * 2π rad/revolution * 1 minute / 60 seconds
= 2.1 * 10^2 * 2π / 60 rad/s
= 7π rad/s (approximately 21.99 rad/s)
Time in seconds:
Time = 68 minutes * 60 seconds/minute
= 4080 seconds
Now, we can substitute the values into the formula to find the average angular acceleration:
Average angular acceleration = (7π rad/s - 160π rad/s) / 4080 seconds
= (-153π rad/s) / 4080 seconds
≈ -0.0373 rad/s²
Therefore, the magnitude of the average angular acceleration of the CD is approximately 0.0373 rad/s².
To find the magnitude of the average angular acceleration of the CD, we can use the formula:
average angular acceleration (α) = (final angular speed (ωf) - initial angular speed (ωi)) / time (t)
First, let's convert the angular speeds from revolutions per minute (rpm) to radians per second (rad/s).
1 revolution = 2π radians
1 minute = 60 seconds
Therefore, we have:
ωi = 4.8 * 10^2 rpm * 2π rad/1 revolution * 1 minute/60 seconds
= 4.8 * (2π/60) rad/s
Similarly,
ωf = 2.1 * 10^2 rpm * 2π rad/1 revolution * 1 minute/60 seconds
= 2.1 * (2π/60) rad/s
Now, let's find the time (t) taken for the CD to go from the initial angular speed to the final angular speed. We are given that the CD has a playing time of 68 minutes. However, we need to convert this time to seconds since we are using rad/s for angular speed.
t = 68 minutes * 60 seconds/1 minute
= 4080 seconds
Now we can substitute the values into the formula:
α = (ωf - ωi) / t
= (2.1 * (2π/60) - 4.8 * (2π/60)) / 4080
= (2.1 - 4.8) * (2π/60) / 4080
= -2.7 * (2π/60) / 4080
= -2.7 * (π/180) rad/s^2
Therefore, the magnitude of the average angular acceleration of the CD is 2.7 * (π/180) rad/s^2.