A flywheel turns through 32 rev as it slows from an angular speed of 2.1 rad/s to a stop. How much time is required for it to complete the first 16 of the 32 revolutions?

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To find the time required for the flywheel to complete the first 16 of the 32 revolutions, we need to determine the average angular speed during that period.

The angular speed (ω) is given in rad/s, and the formula for average angular speed (ω_average) is:

ω_average = (ω_initial + ω_final) / 2

where ω_initial is the initial angular speed and ω_final is the final angular speed.

Given that the initial angular speed (ω_initial) is 2.1 rad/s and the final angular speed is 0 rad/s (since it comes to a stop), we can substitute these values into the formula:

ω_average = (2.1 + 0) / 2
= 1.05 rad/s

We know that the time (t) required to complete a revolution (one full cycle) is given by the formula:

t = 2π / ω_average

where π is the mathematical constant pi (approximately equal to 3.14159).

Substituting the value of ω_average into the formula:

t = 2π / 1.05
≈ 6.01 seconds

Since we want to find the time required for 16 revolutions, we can multiply the time for one revolution by 16:

16 * 6.01 ≈ 96.16 seconds

Therefore, it takes approximately 96.16 seconds for the flywheel to complete the first 16 of the 32 revolutions.