How fast (in rpm) must a centrifuge rotate if a particle 6.0 cm from the axis of rotation is to experience an acceleration of 130000 g's
Ac = v^2/r = r omega^2
so
omega = sqrt (130,000/.06) radians/s
to get rpm from radians/sec
rev/min = omega in rad/s *(1 rev/2 pi rad)(60 s/min)
To determine the required rotational speed of the centrifuge, we can use the following equation:
R = (g * a) / (ω²)
where:
R is the distance of the particle from the axis of rotation (in meters),
g is the acceleration due to gravity (approximated as 9.8 m/s²),
a is the desired acceleration of the particle (in meters per second squared),
and ω is the angular velocity of the centrifuge (in radians per second).
First, let's convert the distance of the particle from the axis of rotation from centimeters to meters:
R = 6.0 cm = 0.06 m
Next, convert the desired acceleration in terms of "g's" to meters per second squared:
130,000 g's = 130,000 * 9.8 m/s²
Now, substituting these values in the equation and solving for ω:
0.06 = (9.8 * 130,000 * 9.8) / (ω²)
Rearrange the equation to solve for ω:
ω² = (9.8 * 130,000 * 9.8) / 0.06
Take the square root of both sides to isolate ω:
ω = √((9.8 * 130,000 * 9.8) / 0.06)
Calculating this equation will give us the angular velocity in radians per second. To convert this to rpm (revolutions per minute), use the formula:
rpm = (ω * 60) / (2π)
Substitute the calculated value of ω into the formula to find the rpm:
rpm = (√((9.8 * 130,000 * 9.8) / 0.06) * 60) / (2π)
This will give you the required rotational speed (in rpm) for the centrifuge to generate an acceleration of 130,000 g's at a distance of 6.0 cm from the axis of rotation.