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What motivates this question is the apparent discrepancy between the concept of Work in physics and in physiology.

Work in physics is defined as the dot product of the force applied to move a certain object by its displacement:

$W = \vec{F} \cdot \vec{d}$

$\text{ Work = (Force) $\cdot$ (Displacement)}$

Where $\vec{F}$ is the force vector, and $\vec{d}$ is the displacement vector.

Now if there is some weight lying on a table, it takes no Work at all to maintain its present configuration, since both $\vec{F}$ and $\vec{d}=0$, As a result $W=0$. In other words, this weight-table system does not need to be provided with energy at all, for the table to be able to support the weight, because it takes no energy to support it.

What about an identical case but with a human hand supporting a weight(at a constant height) at this time? Although according to the above reasoning one should expect that it does not need any energy to support the weight, this is not what we observe in everyday life, because when I hold the weight, it takes a tremendous amount of effort to support and maintain its position.

When I looked up the reason behind this, I found out that it has to do with the fact that the muscles of the arm are skeletal muscles not a smooth one.

If the arm were made out of smooth muscles, then it would take no effort or energy at all to support any weight at a fixed height.On the other hand, the muscle fibers that contract to support the weight in Skeletal muscles has to be continuously provided with energy to be able to support it, and hence although no mechanical work is applied on the weight so $W=0$, nevertheless energy is wasted in the form of heat in this process, and hence we get tired after holding an object for some time.

So what is the mechanism by which Skeletal muscles support certain weight? Why it needs continuous supply of energy while holding an object? How is this energy supplied( my guess is:some component of a cell breaking down molecules so as to transfer chemical energy to mechanical contraction of muscle fibers)?And how it differs from smooth muscles?

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    $\begingroup$ Welcome to Biology and thanks for your question. I think the big difference with a table and a human arm is the presence of joints. Even if holding a weight straight in the air, it is sill a balancing act that needs continuous muscle strain. $\endgroup$ – AliceD Jun 15 '15 at 5:46
  • $\begingroup$ Do you have a reference for your statement about smooth muscle? It seems implausible to me. $\endgroup$ – Roland Jun 15 '15 at 5:56
  • $\begingroup$ @Rolan the Feynman lectures on physics : "The smooth muscles work very slowly, but they can hold a “set”; that is to say, if the clam tries to close its shell in a certain position, it will hold that position, even if there is a very great force trying to change it. It will hold a position under load for hours and hours without getting tired because it is very much like a table holding up a weight, it “sets” into a certain position, and the molecules just lock there temporarily with no work being done, no effort being generated by the clam." $\endgroup$ – Omar Nagib Jun 15 '15 at 6:24
  • $\begingroup$ "Smooth muscle would be much more effective for holding up weights because you could just stand there and it would lock in; there would be no work involved and no energy would be required. However, it has the disadvantage that it is very slow-operating." $\endgroup$ – Omar Nagib Jun 15 '15 at 6:24
  • $\begingroup$ @OmarNagib You should note that even in pure mechanical examples, the energy consumed may not be equal to work done because some energy is also used up in countering the frictional force. $\endgroup$ – WYSIWYG Jun 15 '15 at 7:03
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For skeletal muscles, you'll find a discussion in the Physics SE: https://physics.stackexchange.com/questions/1984/why-does-holding-something-up-cost-energy-while-no-work-is-being-done

To make things short: while holding a position, muscles need to maintain a given length. However, internally, the muscle looks like a collapsible array of filaments, with myosin heads acting as crossbridges between them. Even when the muscle is not contracting, myosin heads unbind and rebind at some rate and would let the muscle relax -- unless you input more energy to compensate this with a further contraction.

For smooth muscle, according to this article, the rate of unbinding can be very slow:

Once the contraction has occurred, the cross bridge attachments need to be released in order that the muscle can relax. Although less is known about this phenomenon, it appears to be related to dephosphorylation of MLC20. In some situations, the dephosphory- lated MLC20 is very slow to allow detachment of the actin from the myosin cross bridge, resulting in a prolonged contraction. This has been referred to as a ‘latch-bridge’ (Hai and Murphy, 1988). This phenomenon may be of great value especially for tonically active SM beds as it would allow them to maintain basal tone through holding in an isometric state without a great energy cost.

So as the muscle is not relaxing, no input of energy is needed to adjust its length to the desired one. I assume a disadvantage of that is that the reaction time of the muscle is slower.

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You should note that even in pure mechanical examples, the energy consumed may not be equal to work done because some energy is also used up in countering the frictional force.

In your example you consider an object kept on a wooden table and in this case the weight is balanced by the reaction force by the table. You have not considered the material properties of the table. The reaction force would be equal to weight only if the weight does not cause any distortion to the object that it is resting on. When distortion happens; for e.g. the if wooden plank bends, the resting object comes closer to ground with a reduction in its potential energy. Now the weight is balanced by the reaction force which is basically the force exerted on the plank to get back to its original position (analogous to the spring; $F=-kx$). Even in this case the material property of the wood is, more or less, invariant.

Getting back to biology. Muscles consume energy in order to maintain their material properties, even if they do not perform work. Muscles can be thought of as dynamic materials because their material properties can change.

When you are holding a dumbbell in your hand with your arm poised at an angle then your biceps need energy to keep maintaining that tension (a temporary material property). The energy is supplied in the form of ATP.

I won't discuss the case of smooth muscles. You should read more about what they exactly do.

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