Well, Pilates is dead, he can't help you in any way.
I have been asked this type of questions many times from prospective students and friends.
So let's analyse the mechanics of fat loss. Allow me to get a bit technical here.
The basic premise that i am taking is that, your training programme must be high intensity to create an environment geared toward fat loss. As mentioned many times elsewhere, this would elevate your metabolic rate for many hours post-workout, giving you a better fat burning effect than conventional long-slow-distance (LSD) cardio.
What does affect intensity? It is the amount of work done in a certain amount of time. In simple physics this is called power.
1) Power = force x velocity = force x displacement / time
2) Power = 0.5 x mass x velocity^2
From these two equations, it is clear without a shadow of doubt that velocity is a component of power, and hence fat loss.
From the second equation, velocity has a factor of two, ie the speed of movement contributes to the power more than the resistance or mass moved.
If you look at a power curve of muscle, you would notice that the biggest area under the curve (ie power), occurs at a moderate level of resistance and velocity.
Image courtesy of http://www.unmc.edu/physiology/Mann/mann14.html
And notice that at maximum resistance and maximum velocity, the area under the curve is ZERO.
Therefore isn't it clear that slow movements or static poses (like in yoga) are not as effective in fat burning? No doubt that muscle contraction by itself requires energy, but mechanically speaking there is no energy consumption if there is no movement, no matter how much the resistance.
The trick is finding the right resistance, this relates to the weight in weight training, leverage or movement difficulty in bodyweight exercise, that allows you to produce the fastest speed of movement (i won't cover this here).
If the speed of movement is fixed, like in a push up, doing more repetitions in the same amount of time would be higher intensity.
Of course there are many other factors involved, but that's it for now.
I hope i have benefitted you from this article. Please post a comment if you have any questions.
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