Semantics, semantics, semantics.
Contacts are the switching components of relays and contactors. Contacts have ratings based upon their ability to switch different types of loads. Motors are difficult loads to switch, so the contacts must be capable of that type of duty or they will not last long. The contacts in the fire alarm are NOT rated for switching any appreciable motor load, so they must switch another device on and off, and that 2nd tier device would be the motor controller, be it a relay or contactor.
A contact
or is just a name given for a relay with heavier duty contacts specifically designed for motor or other difficult load switching. That isn't to say that relay contacts are not motor rated, some are, but that is not their primary design criteria. Relays are typically designed so as to switch very fast and with low power consumption, contactors are not.
Heavy duty itself is a relative term. The relay in the picture IS a heavier duty device compared to the .5A contacts in the fire alarm relay. The relay appears to be an Idec RH2, with contacts that are indeed motor rated for 1/3HP at 240VAC, or 1/6HP at 115VAC. If your motor is that size or below, you can use this relay to switch the motor on and off, you will not need another "contactor". If the motor is larger than that, then don't bother with this relay, go buy a contactor.
Now as to the wiring, you did not post any details on the motor, it will make a difference.
Eng-Tips: Help for your job, not for your homework Read faq731-376
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