A centrifuge in a medical laboratory is used to separate the white blood cells, red blood cells, and plasma. When turning off the centrifuge, it slows down from 4200 rev/min to a stop as it rotates through 47 revolutions. What is the angular acceleration of the centrifuge?
We can use the formula for angular acceleration:
α = (ωf - ωi)/t
where ωi is the initial angular velocity, ωf is the final angular velocity (in this case zero, since the centrifuge comes to a stop), and t is the time it takes to slow down (in this case, the time it takes to rotate through 47 revolutions).
First, we need to convert the initial angular velocity from revolutions per minute (rev/min) to radians per second (rad/s):
ωi = 4200 rev/min * (2π rad/rev) * (1 min/60 s) = 439.8 rad/s
Next, we need to find the time it takes to slow down through 47 revolutions. Since we know the initial and final angular velocities, we can use the formula:
θ = (ωi + ωf)/2 * t
where θ is the angle (in radians) rotated during the slowing down period (in this case, 47 revolutions * 2π radians/revolution = 94π radians).
Solving for t, we get:
t = 2θ/(ωi + ωf) = 188π/439.8 = 1.35 s
Now we can use the formula for angular acceleration with the values we have calculated:
α = (ωf - ωi)/t = -ωi/t = -439.8 rad/s / 1.35 s = -326.7 rad/s^2
(Note that the negative sign simply indicates that the angular acceleration is opposite in direction to the initial angular velocity.) So the angular acceleration of the centrifuge is 326.7 rad/s^2.
To find the angular acceleration of the centrifuge, we need to use the formula:
ω² = ω₀² + 2αθ
Where:
ω = final angular velocity (0 rad/s, as it stops rotating)
ω₀ = initial angular velocity (4200 rev/min)
α = angular acceleration (unknown)
θ = total angle rotated (47 revolutions)
First, we need to convert the initial angular velocity from rev/min to rad/s. Since 1 revolution is equal to 2π radians:
ω₀ = (4200 rev/min) * (2π rad/1 rev) * (1 min/60 s) = 440 π rad/s
Now we can substitute the known values into the formula:
0² = (440 π)² + 2α * 47 rev * (2π rad/1 rev)
Simplifying:
0 = 193600π² + 188πα
To find α, we can rearrange the equation:
α = -193600π² / 188π
Simplifying further:
α = -193600π / 188
Therefore, the angular acceleration of the centrifuge is approximately -3250.81 rad/s².