How fast (in rpm) must a centrifuge rotate if a particle 9.50 cm from the axis of rotation is to experience an acceleration of 113000 g's? Need help please.
w^2 r= 11300*9.8 m/s^2
solve for w ( in rad/sec, change that to rpm)
To calculate the required rotational speed of a centrifuge to achieve a specific acceleration, we can use the equation for centripetal acceleration:
a = (r * ω²)
where:
a is the acceleration,
r is the radius from the axis of rotation to the particle, and
ω (omega) is the angular velocity in radians per second.
In the case of the centrifuge, we know the particle's acceleration (113000 g's) and the radius (9.50 cm). However, we need to convert the acceleration from "g's" to meters per second squared (m/s²) and the radius from centimeters (cm) to meters (m) for consistent units.
1 g = 9.8 m/s²
1 cm = 0.01 m
Converting the acceleration:
a = 113000 g's * 9.8 m/s² / 1 g
a = 1107400 m/s²
Converting the radius:
r = 9.50 cm * 0.01 m/cm
r = 0.095 m
Now we can rearrange the equation to solve for ω:
ω = sqrt(a / r)
Substituting the values into the equation:
ω = sqrt(1107400 m/s² / 0.095 m)
Calculating the square root and simplifying:
ω = 33360.23 rad/s
Since we want the answer in revolutions per minute (rpm), we can convert radians per second to rpm:
1 rpm = 2π radians per minute
ω in rpm = 33360.23 rad/s * 1 min / (2π rad) * 60 s / 1 min
Canceling units and simplifying:
ω in rpm ≈ 317616.62 rpm
Therefore, the centrifuge must rotate at approximately 317,616.62 rpm for a particle 9.50 cm from the axis of rotation to experience an acceleration of 113,000 g's.