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.