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) of the centrifuge filter drum, we can use the following steps:
Step 1: Convert the force from Newtons to grams.
Since the force is given in Newtons per gram (N/g), we need to convert it to Newtons. Given that 1 N = 1000 g, we can convert the force as follows:
6 N/g * 1000 g = 6000 N.
Step 2: Calculate the acceleration.
Using Newton's second law of motion, we can relate force (F) and acceleration (a) as follows:
F = ma,
where F is the force (6000 N) and a is the acceleration. Rearranging the equation, we have:
a = F/m,
where m is the mass.
Step 3: Calculate the acceleration in terms of angular acceleration (ω).
The relationship between linear acceleration (a) and angular acceleration (ω) is given by:
a = ω²r,
where ω is the angular acceleration and r is the radius of the filter drum.
Step 4: Substitute values and solve for ω.
In this case, the radius (r) is given as 350 mm, which can be converted to meters by dividing by 1000:
r = 350 mm / 1000 = 0.35 m.
Substituting the values of a and r into the equation, we get:
6000 = ω² * 0.35.
Solving for ω, we divide both sides of the equation by 0.35 and then take the square root:
ω² = 6000 / 0.35,
ω² = 17142.86,
ω = √(17142.86),
ω ≈ 131.03 rad/s.
Step 5: Convert angular speed to RPM.
To convert the angular speed from radians per second (rad/s) to revolutions per minute (RPM), we can use the following conversion factor:
1 rad/s = (60/2π) RPM.
Substituting the value of ω and calculating, we have:
ω = 131.03 rad/s * (60/2π),
ω ≈ 1239.6 RPM.
Therefore, the required speed of the centrifuge filter drum is approximately 1239.6 RPM.