I found that this small metal square labelled "1/4 FE" had come off my car's wheel yesterday. I am wondering what it could be, and whether it's important to put it back on.
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Those appear to be wheel weights. When mounted on the rims, tires are often not totally and absolutely balanced. When you drive, it may cause all sorts of shaking issues if it is badly out of balance. To compensate, the mounter will spin the tire and get a reading on how to use weights on the rim to compensate for the out-of-balance condition. They are VERY important. Get your tires rebalanced or you will start to wear them more quickly and possibly increase wear on the rest of your wheel system.
This is a wheel balancing weight. This type is held onto the inner side of the wheel with double-backed foam type adhesive tape. The the "1/4" stands for one quarter of an ounce. The "FE" denotes that it is made out of iron. Others are made from zinc and lead. Lead type are banned in some countries and states.
It is important and needs to be replaced if it came off an in use wheel. A wheel balancer machine is needed to get the placement correct. One may be able to discover where it came from because one half of the tape usually stays on the wheel.
Picture of wheel with weights in place and old adhesive from a previous installation. Blue arrow weights. Yellow arrow old adhesive.