This is such a good question and it made me do some googling to figure out the answer.
When you exert a force on the end you’re holding, it doesn’t instantly exert a force on the other end. The atoms have to transmit that force down the length one by one, and that happens at a finite speed. So, even with the most rigid material possible, over a huge distance the rod would kind of look like a palm tree or a bamboo stick being swung – the far end would take some amount of time to “catch up” to where a perfectly straight line would be.
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