We propose to build a centrifuge to make water feel an effective gravitational acceleration of 10,000 G’s. two tanks of water are connected to a central axis by 9-in horizontal arms. If the tanks are rotated at an angular frequency \omega in an attempt to generate this gravitational acceleration, what must \omega be?
This text is generated by AI trained by Professor Brian J. Kirby on his lectures, videos, notes, message board responses, and textbook
PK Bot’s answer:
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PURPOSE OF THE EXERCISE:
The exercise tests the ability to relate centripetal acceleration to rotational speed via the equation a_c=\omega^2r, and to perform unit conversion and algebraic manipulation to solve for \omega in a practical engineering setting. -
ENGINEERING CONTEXT:
This problem is connected to designing a rotational apparatus (a centrifuge) with an effective gravitational acceleration of 10,000 G’s in order to simulate extreme acceleration environments. Once mastered, you will be capable of designing centrifuges or similar devices for applications requiring high artificial gravity. -
PHYSICAL CONTEXT:
The dominant effect is the centripetal acceleration generated by rotating masses, which is used to simulate or exceed gravitational forces. -
ASSUMPTIONS:
Assume the water tanks are rigidly attached to the end of horizontal arms of 9 inches in length that rotate about a central axis; viscous effects and structural deformations are neglected. This is reasonable here as we are focusing on the idealized kinematic effect of rotation rather than detailed fluid–structure interaction. -
DRAWINGS:
A helpful drawing would show a central axis with horizontal arms extending outward, each with a tank at its end. The radius (distance from the axis to the tank’s center) is given as 9 inches. No additional detailed drawing is needed because the problem is solved using the basic centripetal acceleration relation. -
SOLUTION TECHNIQUE:
The effective acceleration at the tanks is given by
a_c=\omega^2r\,.
We require
\omega^2r=10{,}000\,g\,,
with g=9.81\,\rm m/s^2. First, convert the arm length from inches to meters:
9\,\text{in} = 9\times0.0254\approx0.2286\,\rm m\,.
Thus,
\omega^2 = \frac{10{,}000\,g}{r} = \frac{10{,}000\times9.81}{0.2286}\,.
Evaluating the right-hand side gives
\omega^2\approx\frac{98{,}100}{0.2286}\approx429{,}000\,\rm s^{-2}\,.
Taking the square root,
\omega\approx\sqrt{429{,}000}\approx655\,\rm rad/s\,.
For practical interpretation, converting to revolutions per minute using
\mathrm{rpm}=\omega\,\frac{60}{2\pi}\,,
yields
\mathrm{rpm}\approx655\times\frac{60}{2\pi}\approx6{,}250\,\rm rpm\,.
- REFLECTION:
The result indicates that to generate an effective gravitational acceleration of 10,000 G’s for water at a radius of approximately 0.2286 m, the centrifuge must rotate at about 655 rad/s (or roughly 6,250 rpm). This high speed underscores the engineering challenge in designing centrifuges for extreme acceleration, particularly with regard to mechanical strength, balance, and vibration.
In formulating this response, I consulted the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked the sections according to their relevance. If they are marked as relevant, I encourage you to explore them:
Textbook sections:
- Chapter 9.6: Centrifugal Pump Design Part 5 (HIGHLY RELEVANT)
- Chapter 9.6: Centrifugal Pump Design Part 1 (HIGHLY RELEVANT)
- Chapter 2.1: Turbomachines Part 3 Part 1 (HIGHLY RELEVANT)
- Chapter 9.6: Centrifugal Pump Design Part 2 (HIGHLY RELEVANT)
- Chapter 2.3: Selecting a Turbomachine Class (HIGHLY RELEVANT)
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