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So with respect to my motor
– its got all the power I need on its starter winding for what its used for.
Correct, and it’s continuously rated (able to deliver power without needing to stop periodically to cool down.) Except substitute “run winding” for “starter winding”,
– With it rated as 240V it suggests its not dual voltage so ‘over heating’ may be non-use of main winding for years? or possible internal wiring issue?
over-heating is always bad. Either due to being wired incorrectly (missing or wrong capacitor), more likely due to internal damage – burnt insulation, insulation failed due to old age or chafed, stuck centrifugal switch (if fitted)
– Do any of the new facts change the advice you have all helpfully given me?
Not so far. But, we don’t know for sure what sort of motor this is, and there are several possibilities. I don’t understand why the yellow/black winding was disconnected: either the start winding is faulty, in which case rewind or replace it. Or it has a jambed centrifugal switch, which it might be possible to restore or replace. Given this is an elderly US made motor, spares might be unobtainium in the UK.
– Presumably if I wired is as the very helpful diagram, but left out the run? capacitor for the present it should run OK if its going to run (assuming I might need to give it a bit of rotation.
I expect so, but only you are in a position to find out. As the exact type of motor hasn’t been identified, we can only guess. As I said, rope-starting single-phase motors used to be a thing, but it died out because it can catch the operator, and, if he’s too slow, cook the motor. Also a time-waster – much more convenient if the motor starts automatically when switched on. But it’s and option if you want to keep a damaged motor in service: personally, I’d cough up and replace the motor.
– Is it OK that there is no start capacitor? ,
Not really.
– is it OK there is no centrifugal switch to take out the start winding once its rotating?
Maybe, see objections to rope-starting above, but it should work. We don’t know if this motor has a centrifugal switch or not. If present it will be fairly obvious at the back-end of the motor if the casing is removed, and might be visible through the cooling slots:

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Unfortunately, I couldn’t find any information as to the exact type of motor or how it is wired. Closest is this blurred diagram:

Which I think translates to this, but let me emphasise this is more guesswork:

And a problem with the diagram it causes overheating. Could be because the diagram is wrong, or, the motor’s start winding is burnt out.
Mucho confusion about single-phase motors! Clive picked me up for saying I thought you have a “Capacitor Start and Run” motor, If so, my book is wrong! “Capacitor Start and Run” is not the same as “Capacitor Start and Capacitor Run”! Modern parlance may be clearer: what I call “Capacitor Start and Run” is a PCM (Permanent Capacitor Motor),
I prefer to understand and explain, if I can, rather than rely on hazy memory and experience. Electric motors rely on two magnetic fields repelling or attracting at the right time so that a rotor turns. Quite easy to arrange with a DC motor, or a 3-phase AC motor, rather difficult in a single-phase motor. A single electromagnet flips North/South 50 times per second so the rotor doesn’t move, even though the electromagnet induces current in the rotor coil. The two currents are in phase, and unless something unbalances it, the rotor just vibrates slightly and gets hot! By imbalance I mean “shifts the phase / relative timing of two magnetic fields”.
Spinning the rotor mechanically creates the necessary imbalance: once the rotor is moving, the magnet fields are out of phase, and attraction/repulsion occurs. Rotor speed is locked, with some slip, to the AC frequency of the power source, hence 1725rpm @ 60Hz, and 1425rpm @ 50Hz.
There are several ways of creating the imbalance needed to start a single-phase motor, and it may also help to reinforce the imbalance even after the motor is running.
Here’s 4 different examples, note the basic similarity. They all have a run winding and a start winding. The start winding is arranged inside the motor at an angle to the run winding. This creates the necessary imbalance, but exactly how the start winding is managed varies considerably:

Top left: run and start windings both connected permanently, no capacitor or centrifugal switch. Simple, reliable, inefficient, and low torque. The start winding needs a lot of Copper.
Top right: run and start both connected permanently, with the start winding fed by a capacitor. The capacitor provides the phase-shift needed to unbalance the magnetic fields, and maintains it when the motor is turning. The capacitor reduces reliability, but the motor is more efficient, with more torque, and a smaller start winding.
Bottom left: only run is connected permanently. The start winding is connected to line via a centrifugal switch until the motor is reaches about 70% of full speed, after which the start winding is disconnected. Less reliable because of the switch, but the start winding can be much thinner.
Bottom right: this common arrangement provides better torque and reasonable efficiency, the cost is reduced reliability. The run winding is always connected to line. As is the start winding via a capacitor. But this capacitor only trickles power into the start winding to help maintain torque: it is not enough to start the motor. The beefy start capacitor is only connected to the winding by a centrifugal switch whilst the motor accelerates up to speed, then it’s disconnected. The start winding is on the thin side, and powering permanently soon overheats it. This type is unreliable because it depends on two capacitors and a mechanical switch.
Single-phase motors are far from ideal – the design is always a compromise, and they all vibrate.
Though they look similar, different types of single-phase motor are not the same inside. They are each arranged mechanically to optimise the necessary magnetic effects for their intended usage; coil positions, wire gauges, gaps, core metal etc. A mass of devil in the detail and the design maths is well beyond me!
Anyway, in practice, connecting a bottom right type motor as if it were a top left type will probably burn out the start winding. But adding a capacitor to a top left type won’t cause any damage – the motor will just have reduced power and torque. Likewise, using the wrong sized capacitor, at least too small, won’t do any damage, which is probably why Robert suggested 4uF.
If a motor needs capacitors they are usually mounted on the case. Not the law though – some designs put the capacitor in a separate box, or inside the control unit. Without the spec, the best we can do is guess.
These days, a good way of sorting a duff single-phase motor is to replace it with a VFD & 3-phase, or – even better – brushless motor with controller. Chinese sewing machine motors are popular – powerful and cheap. They’re industrial, not to be compared with the dinky Universal motors found on grannies Singer!
Dave