Why is it common to rest between sets of repetition? Could I, instead of resting, exercise another muscle group?
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Exercise depends on several factors:
So depending on the intensity of your exercise, all of these systems will be taxed.
After a certain amount of reps, you have depleted all your muscles local storage of glycogen and they will need to recharge before they're able to perform work again. After the local storage is gone, your blood will have to resupply your muscles to deliver you new energy. This also means, that if you start working on another muscle group, that muscle group has to share its new supplies of energy with the muscles that are recovering.
Even if you were to rest, your body has a storage of lactate acid it needs to process and an oxygen deficit to compensate for. This requires oxygen, but more importantly produces a lot of CO2 your body needs to get rid off. This is called having an oxygen debt. If you switch muscle groups, you don't give your body time to recover from its oxygen deficit.
If you're trying to work out on your maximum lung capacity, you have to subtract a part of your oxygen supply for the oxygen debt, so you're not working out maximally anymore: you're simply out of breath. Furthermore, if you can't supply your muscles with sufficient oxygen, it will anaerobically burn it's fuels. While they're a great fuel, it won't last you very long and it will most likely hamper the amount of repetitions you can make with the same fuel. Because as soon as you rest, then the fuel get's processed again, but you already moved on to another muscle group...
If your body feels tired, your central nervous system (CNS) might also inhibit any further exercise. It won't stop you from trying, but you probably won't be able to perform the amount of reps you could perform with sufficient rest.
So all in all, even if you were to switch muscle groups: your body needs some rest between repetitions. How much rest your body needs depends on: how heavy the exercise is, how fast your performing it and how many times you perform it per repetition.
Without resting, you won't have enough energy to properly break down your muscle tissue so that it can build back up. If your goal is to build muscle, then to get the best results, you need to break down that tissue.
If you're tired from doing other exercises, you won't achieve the most repetitions in your sets.
As Dan John would say, it all depends on your goals. In fact a lot of programming, or how you string your exercises together both throughout a session and throughout a cycle, has to do with your goals. There are a few approaches to programming:
In programming for a lifting competition, a common practice is to have a main lift for the session. The main lift has plain old sets and reps at a set weight, with as much rest as you need to make sure you get the next set. The assistance work is where you get to have all the fun with the training variables.
I'm surprised no one has mentioned supersets, an ancient and honorable practice in which you do a set of one exercise then immediately do a set of another, rest and repeat. Supersetting 3 exercises is not uncommon. One kind of superset is the leg extension - lat pulldown type. It is done for time efficiency since you get to skip a rest period. Is your performance reduced? Probably. For sure if you're not in great shape or not using steroids. But if today's workout is chest and you do all your benching first, so what? The oddest kind of superset is the dumbbell curl - lat pulldown type. Here you're trying to take your biceps out of the back work, or paradoxically, you're hoping the pulldowns will extract the last bit of bicep strength. The most sophisticated superset is something I recently learned from Eric Cressey's Show and Go where an exercise is paired with a stretch or mobility exercise. As you can see, this is an advanced approach in every sense. All your energy systems need to be well developed and you need to know the primary and secondary muscles used by each exercise. If you're asking the question, you probably aren't ready, but the fact that you came up with it shows a creative mind that will take you far.