What angular speed (in revolutions/second) is needed for a centrifuge to produce an acceleration of 1000g at a radius arm of 15.0cm?
r w^2 = 1000 g = 9800 m/s^2
Solve for omega (w) in radians per second. Then convert that to rpm
1 rev/min = 2 pi/60 radians/s
To find the angular speed required for a centrifuge to produce a certain acceleration, we need to use the formula:
a = ω²r
where:
- a is the acceleration
- ω (omega) is the angular speed
- r is the radius arm
In this case, the acceleration is 1000g, and the radius arm is 15.0cm.
1. Convert the radius arm from centimeters to meters:
r = 15.0 cm = 0.15 m
2. Convert the acceleration from g to m/s²:
Since 1g = 9.8 m/s²,
acceleration in m/s² = 1000g * 9.8 m/s²/g = 9800 m/s²
3. Rearrange the formula to solve for angular speed ω:
ω = √(a / r)
Substitute the values:
ω = √(9800 m/s² / 0.15 m)
4. Calculate the square root and divide:
ω = √(9800 m/s²) / √(0.15 m)
ω = 99 m/s / 0.387 m
ω ≈ 255.7 rad/s
To convert the angular speed from radians per second to revolutions per second, we use the conversion factor:
1 revolution = 2π radians
5. Apply the conversion factor:
ω (in revolutions per second) = ω (in radians per second) / (2π)
Substitute the value:
ω (in revolutions per second) ≈ 255.7 rad/s / (2π)
ω (in revolutions per second) ≈ 40.7 rev/s (rounded to one decimal place)
Therefore, the angular speed needed for the centrifuge to produce an acceleration of 1000g at a radius arm of 15.0 cm is approximately 40.7 revolutions per second.
To determine the angular speed needed for the centrifuge, we can use the formula for centripetal acceleration:
a = ω^2 * R
where:
a is the centripetal acceleration
ω is the angular speed
R is the radius arm
Given:
a = 1000g (acceleration)
R = 15.0 cm (radius arm)
We need to convert the given values into SI units:
1g = 9.8 m/s^2
R = 15.0 cm = 0.15 m
Now, we can rearrange the formula to solve for ω:
ω^2 = a / R
ω = √(a / R)
Substituting the values:
ω = √(1000g / 0.15)
Converting g to m/s^2:
ω = √(1000 * 9.8 / 0.15)
Calculating:
ω ≈ √(9800 / 0.15)
ω ≈ √(65333.33)
ω ≈ 255.31 rad/s
To convert the angular speed from radians per second (rad/s) to revolutions per second (rev/s), we divide by 2π:
ω (in rev/s) = ω (in rad/s) / 2π
ω (in rev/s) ≈ 255.31 / (2π)
Calculating:
ω (in rev/s) ≈ 40.66 rev/s
Therefore, the angular speed needed for the centrifuge to produce an acceleration of 1000g at a radius arm of 15.0 cm is approximately 40.66 revolutions per second.