A certain CD has a playing time of 72 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=(vf-vi)/time
change 72min to seconds, change velocities to radians/sec
angacceleration=2PI/60 (480-210)/72*60
To find the magnitude of the average angular acceleration of the CD, we need to determine the change in angular velocity and the time it takes for this change to occur.
Given:
Initial angular velocity (ω₁) = 4.8 x 10² rpm
Final angular velocity (ω₂) = 2.1 x 10² rpm
Step 1: Convert the angular velocities from rpm to rad/s:
1 revolution = 2π radians
So, the initial angular velocity (ω₁) = 4.8 x 10² rpm x 2π rad/1 min = 4.8 x 10² x 2π/60 rad/s
= 4.8 x 2π/60 rad/s = 0.8π rad/s
Similarly, the final angular velocity (ω₂) = 2.1 × 10² x 2π/60 rad/s
= 2.1 x 2π/60 rad/s = 0.7π rad/s
Step 2: Calculate the change in angular velocity (Δω):
Δω = ω₂ - ω₁ = 0.7π rad/s - 0.8π rad/s = -0.1π rad/s
Note: The negative sign indicates a decrease in angular velocity.
Step 3: Find the time taken for this change in angular velocity.
Given the playing time of the CD is 72 minutes, we can convert it to seconds:
Playing time (t) = 72 min x 60 s/min = 4320 s
Step 4: Calculate the average angular acceleration (α):
Average angular acceleration (α) = Δω / t = (-0.1π rad/s) / 4320 s ≈ -3.66 x 10^-5 rad/s²
So, the magnitude of the average angular acceleration of the CD is approximately 3.66 x 10^-5 rad/s².