I'm flying in economy with United Airlines, and I know there are limits on the physical size of carry-on bags, but is there a certain weight limit for them?
No, there is no weight limit. The only limits are dimensions:
Carry-on bag: The maximum dimensions for a carry-on bag are 9 inches x 14 inches x 22 inches (22 cm x 35 cm x 56 cm), including handles and wheels.
Personal item: The maximum dimensions for your personal item, such as a shoulder bag, backpack, laptop bag or other small item, are 9 inches x 10 inches x 17 inches (22 cm x 25 cm x 43 cm).
You can find the full policy HERE.
However, you should note that the overhead luggage bins themselves do have a weight limit. This varies by aircraft. Usually it isn't more than about 50-75 Kg (100-150 LBS).
Unless you're carrying gold bullion, basically, you're fine - they don't have an official weight limit. The reason I mention something extra heavy is that generally, you're meant to be able to lift it unassisted into the overhead compartments (or someone else if you're injured). So too heavy and they're going to have to check it.