You have it correct. The battery is probably does not have enough reserve to transmit.? Not a real problem.? If you can’t get a replacement battery from Yaesu, you can get one from Batteries America or another 3rd party seller.
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There are ways of testing the battery, but if it is old, it is probably not worth the trouble or the cost of an external unit.
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The wall wart will not provide enough current to transmit.? You generally can get a 12 v car adapter to plug into the power socket, and with a 12 V power supply with a 12v socket (or of course, your car) or a 12v to bare wires could be used to connect to a 12 v supply.
I just "inhereted'? an FT-60r from a SK estate. just the radio. Picked up a generic desk charger, charged it up, and tried to hit a local repeater. Radio started fine , but went off, then restarted as soon as i tried to transmit. I think I know the answer to this before I ask. Is it the Battery? It charged awful fast. How to I check the battery? By the way, it exhibits the same behavior when plugging the wall wart into the radio itself. Any help or suggestions will be appreciated. Greatly.