Yes, but not in any meaningful way. The difference in weight fully charged car battery is about the same as the mass of rubber that your tires lose when you accelerate from a stop.
It only increases in mass (and thus weight, all else being equal) thank’s to relativity. Something with more energy is going to have more mass. In the same way that things going faster have more mass. This is simply a non-factor in day-to-day life but it’s *technically* true.
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