" writes:
I have a very old table saw (runs off a pulley) that won't start
unless I give the pulley wheel a gentle spin just as I turn on the
switch to the saw. I'd like not to have to start the saw this way (for
obvious reasons), but don't know enough about electrical motors,
capacitors, centrifugal switches, etc. to know how to fix this
problem. Can anyone out there help me?

Other people have already given you a bunch of good advice about how to
fix the motor. I thought I'd try to explain *why* it behaves this way:

A single-phase induction motor with only one energized winding will not
start rotating on its own. If you give it a push in *either* direction
by hand, it will then accelerate to speed and continue running in the
direction you started it. But it won't start.

To avoid the inconvenience of manual starting, all single-phase motors
have an additional winding of some sort. The second winding is
mechanically rotated from the main winding, and it's fed with current
that is somehow shifted in phase relative to the main winding. This
creates an effectively rotating magnetic field, which makes the motor
always start in one direction. Anything that prevents this second
winding from getting power will also prevent the motor from starting on
its own.

The simplest design mechanically is the split-phase motor. A
centrifugal switch is supposed to apply power to the second winding when
the motor is stopped, but disconnect before it reaches full speed.
Switch contacts can get burned, or the switch just wears out.

A capacitor-start motor uses a capacitor in series with the second
winding, again with a switch that opens when the motor reaches full
speed. The switch can be bad, or the capacitor can be bad as well.

A capacitor-run motor keeps the capacitor and the second winding
connected all the time. Here, the second winding both provides reliable
starting and more running torque. Again, a bad capacitor means it won't
start.

Dave