What angular speed (in revolutions/second) is needed for a centrifuge to produce an acceleration of 1000g at a radius arm of 15.0cm?
To find the angular speed needed for the centrifuge, we first need to understand the relationship between the acceleration, radius, and angular speed.
The acceleration of an object moving in a circle is given by the equation:
a = r * ω^2,
where a is the acceleration, r is the radius, and ω (omega) is the angular speed.
In this case, the acceleration is given as 1000g. To convert from g to meters per second squared (m/s^2), we use the conversion factor of 9.8 m/s^2 = 1g. Therefore, 1000g is equal to 1000 * 9.8 m/s^2.
The radius of the centrifuge arm is given as 15.0 cm. To convert to meters, we divide by 100. So the radius, r, is equal to 15.0 cm / 100 = 0.15 m.
Substituting the given values into the equation, we have:
1000 * 9.8 = 0.15 * ω^2.
Simplifying the equation, we get:
9800 = 0.15 * ω^2.
To solve for ω, we divide both sides of the equation by 0.15:
9800 / 0.15 = ω^2.
Now, take the square root of both sides to solve for ω:
ω = √(9800 / 0.15).
Using a calculator, we find:
ω ≈ 171.5 radians/second.
To convert from radians/second to revolutions/second, we divide by 2π:
ω ≈ 171.5 / (2π) ≈ 27.3 revolutions/second.
Therefore, the angular speed needed for the centrifuge to produce an acceleration of 1000g at a radius arm of 15.0 cm is approximately 27.3 revolutions/second.