H&S look away.
Slip the belt off, wrap around the shaft a length of soft cord, note the direction you did the wrapping, give the cord a firm pull and switch on. Note motor direction of rotation, switch off. Do the same with the cord, but in the other alternative direction, switch back on and again note DOR. You have just tested the run winding for continuity. So your fault is in the start circuit and most likely the centrifugal switch..
That terminal box doesn't look like the original one and no signs of a capacitor, so as Dave says, a spilt phase. Before you take it apart to hopefully locate the fault, with a scriber mark both bell ends. Two marks on the drive end and single on the non drive end straight across, these are to help you when reassembling the motor. When withdrawing the rotor be very care full not to scratch the windings and break a wire, as it can be very difficult to locate the other broken end and carry out a repair, also don't let the rotor fall on the workshop floor and bend the shaft. Do not oil the centrifugal switch mechanism, you might need to clean up the contacts. Once you have the motor reassembled, and for your own personal safety you ought to get your friendly electrician, to give it an insulation test, before connecting to the supply once again Let us know how you get on and don't forget the most important wire the EARTH. John