What makes the burn?
When you exercise, your body breaks down sugars to produce energy for your muscles. If your body is continually getting the oxygen that it needs, the only waste products are carbon dioxide and water. But if you exercise so intensely that your body cannot keep up with the oxygen demands, carbon dioxide is not removed from the cells. The carbon dioxide causes lactic acid to build up in your muscles and eventually flow into your bloodstream. The only way to prevent lactic-acid buildup is to increase your body's ability to utilize and transport oxygen by properly aerobically training your body.
So for growth, NO you don't need to feel it.
I personally use the burn as a guide line in how far I am pushing myself compared to what the body is used to.
They know now muscles recover energy faster than they told people for years.
If you don't feel a burn in a set, they have found in 30sec the muscle is fully replenished. With this sead, if you take longer than 30 sec breaks then you may not reach the burn.
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