The Centrifuge Problem
Today we had a lab practical. I wanted to spin together with another group (save 10 minutes of waiting), so I asked if I could spin with that group. They had 4 people. We had 3. So altogether, that made 7 tubes. I rearranged the 7 tubes in the manner shown in the picture below (Green spots indicate position of tubes). Our centrifuge has 12 holes, each evenly spaced. They insisted that the centrifuge was imbalanced and refused to allow me to spin it in that manner.
My friends also agreed that it was imbalanced. It made me upset, so we decided that we'd ask the prof about it. So we took my scribbling and asked the prof who was right. The prof also agreed that it was imbalanced (i.e. I was wrong.)
I can prove mathematically that I am, in fact, correct.
Since our centrifuge has 12 holes, each evenly spaced, each hole is away from every other hole 360/12=30 degrees on the circle.
Centrifugal forces act outward. Hence each tube can be modelled as a point object with a force facing outwards. As each of our tubes are identical, let's take into account only the direction of the forces for all the tubes. Resolving the forces into the horizontal and vertical directions, we have:
Horizontal direction: sin 0 + sin 30 + sin 120 + sin 150 + cos 210 + sin 240 + sin 330 = 0
Vertical direction: cos 0 + cos 30 + cos 120 + cos 150 + cos 210 + cos 240 + cos 330 = 0
Note that both the horizontal components and vertical components of the forces all sum to zero. This means that my centrifuge is indeed, balanced properly.
Hmph. Idiots who insist that I am wrong. Pfft. Take that!
My friends also agreed that it was imbalanced. It made me upset, so we decided that we'd ask the prof about it. So we took my scribbling and asked the prof who was right. The prof also agreed that it was imbalanced (i.e. I was wrong.)
I can prove mathematically that I am, in fact, correct.
Since our centrifuge has 12 holes, each evenly spaced, each hole is away from every other hole 360/12=30 degrees on the circle.
Centrifugal forces act outward. Hence each tube can be modelled as a point object with a force facing outwards. As each of our tubes are identical, let's take into account only the direction of the forces for all the tubes. Resolving the forces into the horizontal and vertical directions, we have:
Horizontal direction: sin 0 + sin 30 + sin 120 + sin 150 + cos 210 + sin 240 + sin 330 = 0
Vertical direction: cos 0 + cos 30 + cos 120 + cos 150 + cos 210 + cos 240 + cos 330 = 0
Note that both the horizontal components and vertical components of the forces all sum to zero. This means that my centrifuge is indeed, balanced properly.
Hmph. Idiots who insist that I am wrong. Pfft. Take that!