They usually don’t. Most things exposed to radioactivity do not then become radioactive.
Think of it like a microwave. Most things you microwave do not continue to generate heat when you take them out. However, some things do (think of something catching on fire). So, most things that are exposed to radiation do not continue to emit radiation, unless that radiation triggered the object to begin its own radioactive decay
Comments here are good. There’s one thing I don’t see yet which is important to think about are decay chains. There are a lot of nuclides (general term for a nucleus with some.number of protons and some number of neutrons) that decay into another unstable nuclide. That decays and again and again until it gets to a stable element. The most common isotope (a nuclide with the same number of protons, but different neutrons the # of protons determines the element) of Uranium is U-238 which, over a chain of 14 decays, becomes Pb-206, a stable isotope of lead.
So if the first decay launches a piece of radioactive thorium onto your jacket, your jacket now emits radiation because of it. (Or rather, the contaminant does, but a Geiger counter doesn’t care, nor does your health, whether the jacket itself or the contaminants are what’s emitting the radiation… In this specific instance, it’s all alpha and beta decay which is… *relatively* harmless… But I’m not taking any chances)
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