Switch is turned on but pump would not run. shaft turns by hand

possible start run capacitor failure
hard to turn while on indicates bearing failure allowing the armature to touch the field poles

1 or 2 capacitors on this motor?
Sometimes there are two...
One is a "Start" capacitor and if a 2nd one, that one is a "Run" capacitor.

If only 1 capacitor and you have replaced it... if not a faulty capacitor, then your windings are shot?

When I test a motor with this symptom, I will use the "Clock Face" method of checking for "Dead Spots" on the winding/stator:
Once you have removed the shaft cap on the back of the motor, I presume you have done this since that is only way to turn the shaft manually to get it to run in the manner you described. Place a small mark (1/4" long?) on the outside perimeter of the shaft, say at the 12:00 position. See picture attached...
Then flip switch, does it start or hum?
Then rotate shaft until your mark is now at the 1:00 position. Flip power switch, running or humming?
Do this at all clock positions as you turn/rotate shaft until you reach 12:00 once more. If the pump hums at all clock positions completely dead motor or bad capacitor. If a dead spot is at only one clock position, then absolutely, bad motor - either rebuild or replace.
Coincidentally the motor may stop each time on a dead spot, where the motor won't start due to a fried portion of the stator/winding/armature, only a major repair fixes this. The purpose of the test is to place the position of the stator in a different position in relation to the windings. If humming all the way around, then possibly a bad capacitor, because it is not starting in any position. If it starts say at all positions except 4:00 and 9:00 then motor is bad.

Good luck

try when off to remove back cover on motor then try to turn motor shaft by hand then try to turn on might be rusted

C. MOTOR HUMS, BUT DOES NOT START—
Check for:
1. Centrifugal switch stuck in open position.or if capacitator start could be bad capacitator. Possible loose connection.
2. Binding of motor shaft. (something stuck in impeller) Does motor turn free. IF not check for jamed impeller. If it she sp2605x7 you can view a manual here:
http://www.haywardnet.com/products/Manuals/pdfs/manual113.pdf

check for a re-set button somewhere on the motor housing.

You will need to disassemble the pump and check the impeller. remove any debris and try turning the impeller by hand. If it does not move or is dragging, you will need to overhaul the pump. Possibly a shaft seal leak has damaged the motor bearing(s). Replace the shaft seal and motor bearings. If the motor is badly deteriorated, you might need to replace the motor. Note... the impeller turns counterclockwise to loosen. you may need special tools to grip and hold the motor shaft and the impeller to prevent damage to the impeller. If the impeller is seized on the shaft from rust, you may have to break it to get it off. reassemble with teflon lubricant on the shaft threads when reinstalling the impeller. This will prevent seizure later.

Inside the back case on the motor you'll see a white piece with two copper colored arms that have two silver points that almost touch. If these are rusty I've found that the motor won't turn over
. A quick fix is to take some 600+ grit Aluiminum oxide sandpaper and give each of those contact points a brisk sanding to remove any oxidation. This will help the points make a positive contact . Make sure while very obvious that the motor is unplugged first.

sounds like the windings in the motor are shot! but does it have a capacitor? if it does you could simply have a bad starting capacitor so don't let it hum for that will surely scar the windings

It sounds as if motor is spinning and pump is free so the impellar must not be attached to the motor shaft. There is a set screw on the impellar to hold it to shaft or the pump impellar could be cracked. Only way to tell is take the impellar housing off.

what kind of motor or pump? Hayward, pac flo?