3) Joe has a playing time of 74 minutes. When the music starts, the CD rotating at an angular speed of 480 revolutions per minute (rpm). At the end of the music, the CD is rotating at 210 rpm. Find the magnitude of the average angular acceleration of the CD (in rad/s^2)
To find the magnitude of the average angular acceleration of the CD, we can use the formula:
Average angular acceleration = (final angular velocity - initial angular velocity) / time
Here, the initial angular velocity is given as 480 revolutions per minute (rpm) and the final angular velocity is given as 210 rpm. We need to convert these values to radians per second (rad/s) before using them in the formula.
To convert from revolutions per minute to radians per second, we use the conversion factor 2π rad = 1 revolution and 60 seconds = 1 minute.
1 revolution = 2π radians
1 minute = 60 seconds
So, the conversion factor is:
1 revolution per minute = 2π radians per 60 seconds
Now let's calculate the average angular acceleration.
Step 1: Convert initial angular velocity to rad/s:
Initial angular velocity = 480 rpm * 2π radians/60 seconds
= 480 * 2π / 60 rad/s
Step 2: Convert final angular velocity to rad/s:
Final angular velocity = 210 rpm * 2π radians/60 seconds
= 210 * 2π / 60 rad/s
Step 3: Calculate the average angular acceleration:
Average angular acceleration = (final angular velocity - initial angular velocity) / time
Since the time is given as 74 minutes, we need to convert it to seconds:
Time = 74 minutes * 60 seconds/minute
= 74 * 60 seconds
Now substitute the values into the formula:
Average angular acceleration = (Final angular velocity - Initial angular velocity) / Time
Average angular acceleration = (210 * 2π / 60 - 480 * 2π / 60) / (74 * 60)
Now, simplify and calculate the answer.