A centrifuge has an angular velocity of 3,000 rpm, what is the acceleration (in unit of the earth gravity) at a point with a radius of 10 cm

Answer:

[tex]a_{r} = 1006.382g \,\frac{m}{s^{2}}[/tex]

Explanation:

Let suppose that centrifuge is rotating at constant angular speed, which means that resultant acceleration is equal to radial acceleration at given radius, whose formula is:

[tex]a_{r} = \omega^{2}\cdot R[/tex]

Where:

[tex]\omega[/tex] – Angular speed, measured in radians per second.

[tex]R[/tex] – Radius of rotation, measured in meters.

The angular speed is first determined:

[tex]\omega = \frac{\pi}{30}\cdot \dot n[/tex]

Where [tex]\dot n[/tex] is the angular speed, measured in revolutions per minute.

If [tex]\dot n = 3000\,rpm[/tex], the angular speed measured in radians per second is:

If gravitational acceleration is equal to 9.807 meters per square second, then the radial acceleration is equivalent to 1006.382 times the gravitational acceleration. That is:

Answer:[tex]a_{r} = 1006.382g \,\frac{m}{s^{2}}[/tex]

Explanation:Let suppose that centrifuge is rotating at constant angular speed, which means that resultant acceleration is equal to radial acceleration at given radius, whose formula is:

[tex]a_{r} = \omega^{2}\cdot R[/tex]

Where:

[tex]\omega[/tex] – Angular speed, measured in radians per second.

[tex]R[/tex] – Radius of rotation, measured in meters.

The angular speed is first determined:

[tex]\omega = \frac{\pi}{30}\cdot \dot n[/tex]

Where [tex]\dot n[/tex] is the angular speed, measured in revolutions per minute.

If [tex]\dot n = 3000\,rpm[/tex], the angular speed measured in radians per second is:

[tex]\omega = \frac{\pi}{30}\cdot (3000\,rpm)[/tex]

[tex]\omega \approx 314.159\,\frac{rad}{s}[/tex]

Now, if [tex]\omega = 314.159\,\frac{rad}{s}[/tex] and [tex]R = 0.1\,m[/tex], the resultant acceleration is then:

[tex]a_{r} = \left(314.159\,\frac{rad}{s} \right)^{2}\cdot (0.1\,m)[/tex]

[tex]a_{r} = 9869.588\,\frac{m}{s^{2}}[/tex]

If gravitational acceleration is equal to 9.807 meters per square second, then the radial acceleration is equivalent to 1006.382 times the gravitational acceleration. That is:

[tex]a_{r} = 1006.382g \,\frac{m}{s^{2}}[/tex]