A centrifuge is a commonly used tool in laboratory research. It uses forces experienced during rapid rotation to separate substances in liquid or solid media according to particle size and density differences. Mia invests in an ultracentrifuge, which is a centrifuge that works at enormous angular speeds. In order to separate a compound of mass 50 grams, she estimates that she needs to achieve a radial acceleration of 3.5 x 105 g, where g is the gravitational constant 9.8 m/s2. Her test tube will be fixed 2.0 cm from the axis of rotation. What angular speed should Mia set her device at?

Given Information:

Acceleration = a = 3.5*10⁵g

Gravitational constant = g = 9.8 m/s²

Distance from axis of rotation = r = 2.0 cm = 0.02 m

Required Information:

Angular speed = ω = ?

Answer:

Angular speed = ω = 1.31*10⁴ rad/s

Explanation:

We know that the centripetal acceleration is given by

a = v²/r

The relationship between linear and angular speed is given by

v = ωr

Where ω is the angular speed

So the acceleration becomes

a = (ωr) ²/r

a = ω²r²/r

a = ω²r

Re-arrange for ω

ω² = a/r

ω = √ (a/r)

The acceleration is

a = 3.5*10⁵g

where g is the gravitational constant

a = 3.5*10⁵*9.81

a = 3.434*10⁶ m/s²

Finally, the required angular speed is

ω = √ (3.434*10⁶/0.02)

ω = 1.31*10⁴ rad/s