How fast (in rpm) must a centrifuge rotate if a particle 8.55 cm from the axis of rotation is to experience an acceleration of 1000 g's? (use 9.8 m/s^2) You may assume constant angular velocity.

To find the required rotation speed of the centrifuge in rpm, we need to use the formula for centripetal acceleration:

a = ω^2 * r

Where:
a is the centripetal acceleration
ω is the angular velocity in radians per second
r is the radius of rotation

First, we need to convert the given acceleration from g's to m/s^2. Since 1 g = 9.8 m/s^2:

a = 1000 * 9.8 m/s^2
a = 9800 m/s^2

Next, we need to convert the radius from centimeters to meters:

r = 8.55 cm
r = 0.0855 m

Now we can rearrange the formula to solve for ω:

ω^2 = a / r
ω^2 = 9800 m/s^2 / 0.0855 m

Taking the square root of both sides to solve for ω:

ω = √(9800 m/s^2 / 0.0855 m)
ω ≈ 329.24 rad/s

Finally, to convert from radians per second to revolutions per minute (rpm), we can use the following conversion:

1 revolution = 2π radians
1 minute = 60 seconds

So, to convert from rad/s to rpm, we can use the following conversion factor:

1 rad/s = (1 revolution / 2π radians) * (1 minute / 60 seconds)

Now, let's calculate the rotation speed in rpm:

ω = 329.24 rad/s
ω (in rpm) = (329.24 * 1 revolution / 2π radians) * (1 minute / 60 seconds)
ω (in rpm) ≈ 3292.4 rpm

Therefore, the centrifuge must rotate at approximately 3292.4 rpm for the 8.55 cm particle to experience an acceleration of 1000 g's.