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Even spacecraft that are designed and optimized for a single mission need to be carefully balanced experimentally, and have ballast added before flight.

The Space Shuttle was reusable and had a comparatively short turn-around time, each time with a different payload.

Was the shuttle's mass distribution or at least its moments of inertia measured pre-flight? If so was ballast ever added then to balance it?

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Not sure what you mean by "experimentally" but the final step before moving the Orbiter out of the Orbiter Processing Facility (OPF) was to weigh it.

The final step in OPF orbiter processing is weighing the orbiter and determining its center of gravity. Vehicle performance is affected by both the orbiter’s weight and its center of gravity, and flight programming requires accurate measurements.

Space Shuttle Orbiter Processing Fact Sheet

Ballast was added or removed to the Orbiter during OPF processing based on the calculated vs. desired mass properties. Ballast boxes were located in the nose gear wheel well and just in front of the body flap aerosurface hinge.

enter image description here

(personal notes, annotated by me)

If the weight and balance as measured was different from that predicted, I am sure they would have done something about it.

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    $\begingroup$ Okay, they measure the "monopole moment" experimentally rather than theoretically, which would be to add up a long list of masses of every object. $\endgroup$ – uhoh Apr 28 at 1:19
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    $\begingroup$ @uhoh, yeah, they did both. There was a long document called the Vehicle Summary Weight Statement (you can see some in this book archive.org/details/nasa_techdoc_19830069020/page/n1/mode/2up) that listed what everything was supposed to weigh. And then the actual measurement in the OPF. $\endgroup$ – Organic Marble Apr 28 at 2:33
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    $\begingroup$ @uhoh "mass properties" is a generic term for weight, moments of inertia, etc. Unfortunately the only reference I know of for how this weighing was actually done costs $20 and I'm not that into it. sawe.org/papers/1559/buy Pity, that. $\endgroup$ – Organic Marble Apr 28 at 21:31
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    $\begingroup$ Yes it seems so. I was hoping there might be a hint of a comparison to "known values" measured on the ground. It's at least interesting to read in terms of the experiment and microgravity effects (gravity gradients, drag, etc.). $\endgroup$ – uhoh Apr 28 at 21:49
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    $\begingroup$ the author of your link is Fraley, J. G. and that name along with the term "mass properties" is found on original page number 272 of this Engineering Innovations article: nasa.gov/centers/johnson/pdf/… $\endgroup$ – uhoh Apr 28 at 21:56

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