A centrifuge for separation of sugar crystals in molasses must exert a force of 6 N/g. If the filter drum has a radius of 350 mm, determine the required speed in the revolution per minute
I tried the following
F = 6 = 10^-3*a
a = 6000 = ω^2*r
To determine the required speed in revolutions per minute (rpm), we need to rearrange the equation for acceleration.
Given:
Force (F) = 6 N/g = 6 N * 9.81 m/s^2 (acceleration due to gravity)
We can use the formula for centripetal acceleration:
a = ω^2 * r
Where:
a = acceleration
ω = angular velocity (in radians per second)
r = radius
Plugging in the values we have:
6000 = ω^2 * 0.35 m (converting the radius from mm to meters)
Now we can solve for ω:
ω^2 = 6000 / 0.35
ω^2 = 17142.857
Taking the square root of both sides:
ω ≈ √17142.857
ω ≈ 130.92 rad/s
Finally, we can convert the angular velocity to revolutions per minute (rpm):
1 revolution = 2π radians
1 minute = 60 seconds
So, the conversion factor is:
1 rad/s = (1 revolution / 2π) * (1/60) rpm
Plugging in the value for ω:
ω ≈ 130.92 * ((1 revolution / 2π) * (1/60)) rpm
ω ≈ 393.46 rpm
Therefore, the required speed is approximately 393.46 rpm.
To calculate the required speed in revolutions per minute (RPM), we need to use the formula that relates the acceleration, angular velocity, and radius of the centrifuge:
a = ω^2 * r
where:
a = acceleration
ω = angular velocity (in radians per second)
r = radius (in meters)
Given that the force exerted by the centrifuge is 6 N/g, and the radius of the filter drum is 350 mm (or 0.35 meters), we can convert the force into acceleration:
F = 6 N/g
1 N = 1 kg * m/s^2
1 g = 9.8 m/s^2
So, 6 N/g = 6 kg * 9.8 m/s^2 = 58.8 m/s^2
Now we can substitute the values into the equation and solve for ω:
58.8 m/s^2 = ω^2 * 0.35 m
ω^2 = 58.8 m/s^2 / 0.35 m
ω^2 = 168 m^2/s^2
ω = √(168 m^2/s^2) ≈ 12.96 m/s (rounded to two decimal places)
To convert the angular velocity from radians per second to revolutions per minute, we multiply ω by the conversion factor:
1 revolution = 2π radians
1 minute = 60 seconds
So the speed in RPM is calculated as:
ω (in rev/min) = (12.96 m/s) * (1 rev / (2π rad)) * (60 s / 1 min)
ω ≈ 393.37 rev/min (rounded to two decimal places)
Therefore, the required speed for the centrifuge is approximately 393.37 revolutions per minute (rev/min).