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Thanks for reaching out! This looks like a piece of an older style rod balance, but it is too damaged to identify the exact replacement you'll need. All of the important components are missing that would help to identify the correct replacement.
Are you able to provide a photo of an undamaged balance removed from the window? If that's not possible then we will not be able to narrow down the list of possibilities. Let me know if you have any questions.
Thanks for the photos. I am confident we have a replacement balance we can recommend for you but we'll need a bit more information than these balances can provided.
To start, you will still need to locate an undamaged balance so that you are able to get the right balance length needed for your window. Then you'll need to obtain the exact weight of the sash so we can confirm the stamp needed according to the weight ranges for each available stamp option.
You'll also need to confirm that the diameter of the balance tube is 1/2", if it is not please let me know.
Take a look at the video below that details all of the information needed to identify the correct replacement balances.