A centrifuge is a common laboratory instrument that separates components of differing densities in solution. This is accomplished by spinning a sample around in a circle with a large angular speed. Suppose that after a centrifuge in a medical laboratory is turned off, it continues to rotate with a constant angular deceleration for 10.4 seconds before coming to rest. If its initial speed was 3870 rpm, what is the magnitude of its angular deceleration? How many revolutions did the centrifuge complete after being turned off?

To find the magnitude of the angular deceleration, we can use the following equation:

ω = ω₀ + αt

Here, ω represents the final angular velocity, ω₀ is the initial angular velocity, α is the angular deceleration, and t is the time.

Given:
ω₀ = 3870 rpm (initial angular velocity)
t = 10.4 seconds (time taken to come to rest)
ω = 0 rpm (final angular velocity since it comes to rest)

First, let's convert the initial angular velocity from rpm to rad/s. Since 1 rpm is equal to (2π/60) rad/s, we have:

ω₀ = 3870 rpm * (2π/60) rad/s = 2π * 3870 / 60 rad/s

Now, we can substitute the given values into the equation:

0 = (2π * 3870 / 60) + α * 10.4

Simplifying the equation:

0 = (2π * 3870 / 60) + 10.4α

Rearranging the equation to isolate α:

-10.4α = (2π * 3870 / 60)

Dividing both sides by -10.4 to solve for α:

α = (2π * 3870 / 60) / -10.4

Calculating α:

α ≈ -1.465 rad/s² (rounded to three decimal places)

Therefore, the magnitude of the angular deceleration is approximately 1.465 rad/s².

To find the number of revolutions completed by the centrifuge after being turned off, we can use the equation:

θ = ω₀t + 0.5αt²

Here, θ represents the total angular displacement.

Since the centrifuge comes to rest, its final angular velocity is 0 rpm, and we can ignore the 0.5αt² term.

θ = ω₀t = (2π * 3870 / 60) * 10.4

Calculating θ:

θ ≈ (2π * 3870 / 60) * 10.4 ≈ 4256.453 rad (rounded to three decimal places)

Now, we can convert the total angular displacement from radians to revolutions. Since 2π radians is equal to one revolution, we have:

Number of revolutions = θ / 2π ≈ 4256.453 / 2π

Calculating the number of revolutions:

Number of revolutions ≈ 677.397 revolutions (rounded to three decimal places)

Therefore, the centrifuge completed approximately 677.397 revolutions after being turned off.