When ac compressor engauges you can notice strain on engine . Shortly after that a line came off condenser with a loud bang.The line that broke goes from compressor to condensor. Now compressor is noisy. What could have caused this?

Make sure that your drive belt and ac belts are tight. A loose belt will look like many problems. Also could be your ac compressor. They have ac compressor lubrication in a can just like the freon. You can notice the extra strain the ac puts on the engine when you turn it on and especially at idle. So 1. Make sure belts are tight and in good condition, if not replace. 2. Check freon level in ac. If freon level is okay add ac oil conditioner.

The condensation will be mostly forming on the suction line to the compressor . It is the moisture in the air forming water droplets when the air hits the cooler pipe. The refrigerant boils off inside the line because of the low pressure cause by the sucking action of the compressor and this boiling off reduces the temperature of the line and so the condensation forms . It will be the larger of the 2 pipes and if this condensation is causing concern then there is a form tube that you can get from the rubbers shops that you can wrap around the line and this insulates the line and so the condensation will not form. When the system is working properly you will notice that the smaller line will be quite hot as the compressor compresses the refrigerant back into a liquid so that at the TX valve in the evaporator(inside the car) releases the liquid into the vacuum created by the compressor sucking on that line. The heat in the car helps boil off the liquid to gas and in doing so removes the heat from inside the car . As it is doing this there is condensation forming on the evaporator fins that drains off out a tube to outside under the car ,hence the pool of water on the road as you drive off.

The line from the compressor goes to the top and the bottom line goes to the evaporator in the car. There should be a receiver/drier before the condenser that takes out any impurities in the gas including moisture from the air which damages the internals of an ac system. The line from the compressor goes to it ---from there to the condenser and the bottom line back to the evaporator. If you do not have a receiver/drier then get one.

condensor has 2 lines,one coming from compressor ,one going to evaporator.

The A/C compressor is the accessory that puts the most strain on the engine. But unless you have a wimpy four cylinder car, it should not really be noticable too much.
Has your A/C been serviced or someone add refrigerant to it? Perhaps an over charge. Is it making noise?

This can be due to noise on the belt the idler pulley or the compressor. When the AC is switched on the strain of the compressor can cause this noise. Also if the bearings of the alternator or the water pump is faulty this can cause the noise when the ac belt is loaded.
finally a strain within the engine when the engine is loaded can be due to setting within the engine.

I would think that it was more likely that some wires shorted due to the strain caused by the broken motor mount and that short circuit is what burned the clutch coil out. With the coil not energizing, the electromagnet that pulls the clutch together and engages it, will not function. The pulley is turning, but the compressor isn't. The end result is if the compressor is not turning, no AC.

the codensor is in front of the radiator and looks like a thin radiator. the compressor is the pump and is located on the engine and is turned by the surpintine belt.

it's not suposed too for several minutes this keeps you from starting it with load of ac compressor on

if it does not come back after a few minutes ck refrigerant pressure at low side port (static no compressor engauged) it's gotta be over 10 psi or press switch thinks it's M/T and won't engauge compressor
add refridgerant

Sounds like the clutch in the ac compressor is bad. Locking up and putting strain on the motor enough to bog it down and kill it.