A phonograph record has an initial angular speed of 37 rev/min. The record slows to 10rev/min in 1.3 s. What is the record’s average angular acceleration during this time interval
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To find the average angular acceleration of the record during this time interval, we can use the formula:
Average Angular Acceleration (α) = (Final Angular Speed (ωf) - Initial Angular Speed (ωi)) / Time (t)
Given:
Initial Angular Speed (ωi) = 37 rev/min
Final Angular Speed (ωf) = 10 rev/min
Time (t) = 1.3 s
First, we convert the angular speeds from rev/min to radians/s:
1 rev = 2π radians
1 min = 60 s
Initial Angular Speed (ωi) = 37 rev/min * (2π radians/1 rev) * (1/60 min/1 s) = 37 * 2π / 60 radians/s
Final Angular Speed (ωf) = 10 rev/min * (2π radians/1 rev) * (1/60 min/1 s) = 10 * 2π / 60 radians/s
Now, we can calculate the average angular acceleration:
Average Angular Acceleration (α) = (Final Angular Speed (ωf) - Initial Angular Speed (ωi)) / Time (t)
= (10 * 2π / 60 - 37 * 2π / 60) / 1.3
= (20π/60 - 37π/60) / 1.3
= -17π/60 / 1.3
= -17π / (60 * 1.3)
≈ -0.212 radians/s^2
Therefore, the record's average angular acceleration during this time interval is approximately -0.212 radians/s^2.
To find the record's average angular acceleration, we can use the formula:
Angular acceleration (α) = (final angular speed - initial angular speed) / time
Given:
Initial angular speed (ω₁) = 37 rev/min
Final angular speed (ω₂) = 10 rev/min
Time (t) = 1.3 s
First, let's convert the angular speeds to radians per second, as it is the standard unit for angular speed:
1 revolution = 2π radians
1 minute = 60 seconds
ω₁ = 37 rev/min * (2π radians / 1 rev) * (1 min / 60 s) = 37 * 2π / 60 radians/s
Similarly,
ω₂ = 10 rev/min * (2π radians / 1 rev) * (1 min / 60 s) = 10 * 2π / 60 radians/s
Now, we can substitute these values into the formula:
α = (ω₂ - ω₁) / t
= (10 * 2π / 60 radians/s - 37 * 2π / 60 radians/s) / 1.3 s
= (20π/60 - 37π/60) / 1.3 s
= (-17π/60) / 1.3 s
≈ -0.425 radians/s²
Therefore, the record's average angular acceleration during this time interval is approximately -0.425 radians/s². The negative sign indicates that the record is slowing down.