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From Wikipedia's Project Vanguard; Launch_history I found some candidates for satellites that had pages which I could search for the term "RPM". This is not an exhaustive search, it's just meant get things started:

And yet a documentary film circa 1960 mentions 150 RPM with respect to Vanguard satellites. From 1957-1959 IBM 704 Computer - Vanguard Satellite Program "Science in Space" (Burroughs Datatron) NASA (alternate version of the video here)

At this point it is traveling parallel to the Earth's surface. However the vehicle is traveling only nine thousand miles an hour, not yet fast enough to overcome the pull of gravity and stay in orbit.

Inside the second stage, the third stage now begins to play its part. Small rockets start it spinning on a turntable. **When the spinning reaches 150 revolutions per minute, the second stage is separated, its job completed. Stabilized by its spin, the third stage fires and gathers speed.

Question: Which (if any) Vanguard satellite spun at 150 RPM? How fast did the rest spin, or at leased would have spun had they succeeded?

Also, earlier in the film:

Laboratory tests helped to insure that the payload will withstand not only the intense vibration of launching and flight, but also in some cases the strain caused by hundreds of revolutions a minute.

Screenshot (annotated) showing rockets used to spin up a Vanguard satellite before the third stage ignites and separates from the second stage, from "1957-1959 IBM 704 Computer - Vanguard Satellite Program "Science in Space" (Burroughs Datatron) NASA" https://youtu.be/IFTsGvRPBFY

above: Screenshot (annotated) showing rockets used to spin up a Vanguard satellite before the third stage ignites and separates from the second stage, from "1957-1959 IBM 704 Computer - Vanguard Satellite Program "Science in Space" (Burroughs Datatron) NASA" below: Screenshot showing a Vanguard-like satellite being spin-tested from the same video:

Screenshot showing a Vanguard-like satellite being spin-tested, from "1957-1959 IBM 704 Computer - Vanguard Satellite Program "Science in Space" (Burroughs Datatron) NASA" https://youtu.be/IFTsGvRPBFY

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Question: Which (if any) Vanguard satellite spun at 150 RPM?

Vanguard 3 (III), SLV-7, 1959

How fast did the rest spin, or at least would have spun had they succeeded?

The design of the third stage was that it was supposed to spin up to about 178rpm, with the release of the satellite timed to happen when the appropriate spin rate is reached.

Vanguard 1 was about 25rpm but experienced up to about 30rpm increase due to 3rd stage burning unexpended fuel.

Vanguard 2 achieved about 98rpm but was bumped by the third stage resulting in -1rpm spin and 15rpm about its equator, resulting in a tumble.

Vanguard 3 used an entirely different third stage that worked as predicted. Spin was designed for 150rpm.

They left the third stage attached to the payload since with fuel expended and it being of lightweight construction was not expected to cause an issue.

https://static1.squarespace.com/static/568b2be99cadb6f998fff0f1/t/5f6d2a4463d90a352ae8505b/1600989796559/Vanguard+Engineering+Summary.pdf

https://history.nasa.gov/SP-4202.pdf

https://www.jhuapl.edu/Content/techdigest/pdf/APL-V05-N01/APL-05-01-Fischell.pdf

https://www.google.co.uk/books/edition/Vanguard_Satellite_Spin_reduction_Mechan/wzwkAQAAIAAJ?hl=en&gbpv=0

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    $\begingroup$ I started writing an answer, but I got interested in the spin mechanism. It seems that the spin table accelerated but not the satellite, then they were coupled together later. I want to look into that whole thing. I think the details are in that "engineering" document you linked, that's what I've been reading. $\endgroup$ Jan 29 at 19:21
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    $\begingroup$ yes, it is interesting, they left the 3rd stage attached to the payload. You can put a better answer than me, I'm supposed to be out of the door by now.. $\endgroup$ Jan 29 at 19:22
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    $\begingroup$ Just looking into it for my own interest. $\endgroup$ Jan 29 at 19:23

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