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I'm very interested in vinyl records and analog music, and the belt of my turntable got loose. Upon such situation it piqued my curiosity, what is the output signal at the end of the arm cartridge wires for a known waveshape if the speed is not the correct one.

Say the record was mean to play sin(wt), a pure sine wave, at 33rpm, then, because of a loosen belt or any other reason, it rotates at a different RPM, how to calculate the changes in such sine wave?

I'm not considering the filters that the needle might apply on the signal, whether it is a low pass, band pass, or high pass nor any other impedances that might alter the signal in any circumstance, just a supposedly ideal needle and cartridge.

Say the record was mean to play sin(wt), a pure sine wave, at 33rpm, then, because of a loosen belt or any other reason, it rotates at a different RPM, how to calculate the changes in such sine wave?

The pitch and tempo will change in proportion to the speed change. At 33 RPM it would already be musically flat as the correct speed is 33¹_/3 RPM. A 1 kHz test tone - common on test records - would, at 33 RPM, give off $ frac 3333.33 text kHz $.

The sinewave would remain a sinewave but stretched in time and, therefore, a lower pitch.

Changing the speed of the platter simply affects how fast the groove is moving under the needle, nothing else.

A sine wave with the time axis compressed or expanded is still a sine wave. In fact, since the groove is a direct mechanical representation of the original complex waveform, you still get the same waveform simply compressed or expanded in time.

To really simplify, a record has wiggles in the groove that correspond to the recorded sound pressure. (This ignores stereo, and any companding, but it answers your question).

Events are recorded onto that wiggly grove as they happen -- you can think of the groove as a picture of the sound, with the time domain turned into events happening as the needle follows the groove.

If you play the record slower, all the events happen more slowly -- the singer sings slower and deeper, the orchestra does too, etc. Speeding it up does the opposite -- a normal recording, sped up, sounds like a hyperactive chipmunk.

Grooves are cut with frequency correction according to RIAA equalization. Playing the record off with wrong speed increases all frequencies by the same factor (corresponding to a shift left/right on the frequency axis of the doubly logarithmic transfer function diagram). Since the frequency correction is not a straight line, this does not just result in a frequency shift but also in an uneven frequency response due to recording and replaying correction no longer being proper inverses.

In addition, the equalization is done in order to reduce excessive signal amplitudes on stylus and pickup. Counteracting this by wrong speed may lead to either excessive amplitudes (electrical or mechanical) or too low signals overlaid with a relatively higher noise floor.

The output voltage of record cartridges is given at 1KHz at some standard velocity of the needle.

Often 5 millivolts output at 5 centimeters per second needle movement. This would be for MOVNG-MAGNET, with thousands of ohms because of the very tiny wires in the FIXED coil.
The high resistance causes a high random thermal electron noise floor.

The lowest noise cartridges are MOVING_COIL, often with resistance under 10 ohms.

But serious amplification, at low noise, and low VDD trash injection, and thorough shielding, is needed. The output voltage is often 0.2 millivolts or even less,
at that stated needle velocity, at 1KHz.

[edit] Some RIAA preamplifiers use common-source JFET amplifiers, and those type of gain stages have ZERO power supply rejection. Thus large RC filters are needed for VDD inputs, and regulators are SHUNT designs, with approximately 1nanoVolt/rootHertz noise density.

I have practical experience with this--record players with variable speed drives used to exist. These were specialty systems intended for blind people--they allowed the listener to speed up the records. They were made variable because not everyone wanted the same speed.

Obviously, for music this would be insane but these units were intended for playing voice--magazines read aloud onto special 8 1/3 rpm 9" flexible plastic records. They were not durable at all (but neither are magazines) but did their job at a much lower cost than other technologies of the day. Other than the variable speed drive, the low speed settings (their highest was 33 1/3), and the ability to survive being mailed as is they were ordinary players.