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I read the definition of work as $$W ~=~ \vec{F} \cdot \vec{d}$$ $$\text{ Work = (Force) $\cdot$ (Distance)}.$$
If a book is there on the table, no work is done as no distance is covered. If I hold up a book in my hand and my arm is stretched, if no work is being done, where is my energy going?

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    $\begingroup$ To think how "biochem" energy is disconnected from the "newtonian" energy in this question think about you getting a fly from NY to Paris. For a few hours your body gained a lot of gravitational and kinetic energy but you still get fatiged and hungry in mid flyght over ocean. At the end a lot of work was applyed to move your body (you can even calculate de potency) thousand kilometers but your body don't gained one "biochem" calorie from the experince $\endgroup$ – jean Oct 19 '15 at 16:42
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    $\begingroup$ Also, something worth adding: no work is being done on the book! The book stays still, at constant gravitational potential. All the energy is expended in your body. $\endgroup$ – Andrea Jan 15 '16 at 21:38
  • $\begingroup$ As someone else has already pointed out, the answer is simple. It is all about entropy of your body. To keep your body organized and overcome the thermodynamic first law, which says the entropy of an isolated system tends to become maximum, you need to take in some energy. These energy finally dissipate into the environment from your body in the form of heat. $\endgroup$ – Zhiqiang Wang Jul 27 '16 at 17:24
  • $\begingroup$ @ZhiqiangWang This is unrelated. You are talking about maintaining coherent metabolism. This about how is the energy expended by the body to maintain is dissipated. $\endgroup$ – G. Bergeron Dec 8 '16 at 10:29
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    $\begingroup$ Feynman discussed that feynmanlectures.caltech.edu/I_14.html $\endgroup$ – lalala Mar 11 '18 at 12:41

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While you do spend some body energy to keep the book lifted, it's important to differentiate it from physical effort. They are connected but are not the same. Physical effort depends not only on how much energy is spent, but also on how energy is spent.

Holding a book in a stretched arm requires a lot of physical effort, but it doesn't take that much energy.

  • In the ideal case, if you manage to hold your arm perfectly steady, and your muscle cells managed to stay contracted without requiring energy input, there wouldn't be any energy spent at all because there wouldn't be any distance moved.

  • On real scenarios, however, you do spend (chemical) energy stored within your body, but where is it spent? It is spent on a cellular level. Muscles are made with filaments which can slide relative to one another, these filaments are connected by molecules called myosin, which use up energy to move along the filaments but detach at time intervals to let them slide. When you keep your arm in position, myosins hold the filaments in position, but when one of them detaches other myosins have to make up for the slight relaxation locally. Chemical energy stored within your body is released by the cell as both work and heat.*

Both on the ideal and the real scenarios we are talking about the physical definition of energy. On your consideration, you ignore the movement of muscle cells, so you're considering the ideal case. A careful analysis of the real case leads to the conclusion that work is done and heat is released, even though the arm itself isn't moving.

* Ultimately, the work done by the cells is actually done on other cells, which eventually dissipates into heat due to friction and non-elasticity. So all the energy you spend is invested in keeping the muscle tension and eventually dissipated as heat.

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    $\begingroup$ Just like real scenario , what happens in solid objects like table. From where does the molecules get energy to remain in straight and hold up the weight of book $\endgroup$ – SMUsamaShah Dec 16 '10 at 20:32
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    $\begingroup$ For the table, the situation is different, because the molecules of the table aren't constantly "relaxing" and "contracting". Once you place the book onto the table, the atoms are pushed in a little bit (depending on how sturdy the table is) and settle into a new equilibrium due to electromagnetic and nuclear forces. $\endgroup$ – Lagerbaer Dec 16 '10 at 22:42
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    $\begingroup$ Unlike your arm, the table does not need to spend energy to hold up the book. It is a completely immobile system, as in: the table is a rigid object that needs no energy to stay still. You were correct when you said that you need to spend some energy to keep the book lifted, but that's because of how your muscles worked. You are not correct in saying that the table needs energy to keep the book lifted. All the table needs to do is apply Force, but applying force does not necessarily mean spending energy. $\endgroup$ – Malabarba Dec 16 '10 at 22:45
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    $\begingroup$ Let me just add that this is exactly the reason why rock climbers on steep walls must try to have their arms straight: Rather than hanging on bent muscles, which costs lots of energy, they hang on their body structure. Analog to the arm/table example. $\endgroup$ – Lagerbaer Aug 11 '11 at 3:29
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    $\begingroup$ This is not quite true! Physical effort is a subjective feeling only marginally related to the amount of energy expended. Holding a book would require almost as much power as slowly lifting it. There might not be mechanical energy produced but consider the case of hovering using air jets: this requires huge of energy. $\endgroup$ – G. Bergeron Dec 8 '16 at 10:30
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This is about how your muscles work -- the're an ensemble of small elements that, triggered by a signal from nerves, use chemical energy to go from less energetical long state to more energetical short one. Yet, this obviously is not permanent and there is spontaneous come back, that must be compensated by another trigger. This way there are numerous streches and releases that in sum gives small oscillations that create macroscopic work on the weight.

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Perhaps an analogy is in order. Lets hold up the book by using an electromagnet (say we put a piece if steel under it ). If the coils were made of superconducting material it would take no energy input to maintain the position/field strength. But if we use ordinary wire, ohmic loses within the coil must be made up for by externally supplied electrical energy.

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The reason is that you need to spend energy to keep muscle stretched.

The first thing you need know is that the work $W=F \Delta x$ is the energy transfer between objects. Hence, there are no work done on the book when it is put on the table because there are no movement.

When your arm muscle is stretched, however, it consumes energy continuously to keep this state so you feel tire very fast. This energy comes from the chemical energy in your body and most of them are converted into heat and lost to the surrounding. In this situation, no energy is transferred to the book, so no work is done.

You can feel the different energy consumption when your arm is stretched in different angle. A particular case is that you put the book on your leg when you sit on a chair so your muscle is relaxed and the energy spent is less.

There are also a special type of muscle, smooth muscle, requires very little energy to keep its state so that it can always keep it stretched and you won't get tire:

Tonic smooth muscle contracts and relaxes slowly and exhibits force maintenance such as vascular smooth muscle. Force maintenance is the maintaining of a contraction for a prolonged time with little energy utilization.

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    $\begingroup$ The default state of muscle is relaxed (i.e. long), and the active state is contracted (i.e. shortened). Remaining relaxed does not require energy beyond that to keep the cells alive; contracting does. $\endgroup$ – dmckee Jan 29 '11 at 17:51
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When contracted, the sarcomeres, the structure that actually do the work in a muscle, take turns doing the work. Only a third of them are engaged at any given moment.

This is because the sarcomere pumps blood as it contracts and relaxes, enabling it to get the energy it needs to do its work for longer periods. The temporary, superhuman strength some people experience may be some sort of override of this normal level of engagement.

This system doesn't have a different mechanism for holding a position, so the same thing goes on when trying to hold an object steady.

But if the muscle is contracted for a very long time and the energy in the blood being pumped becomes insufficient, sarcomeres will actually get stuck in their contracted position. This state doesn't require energy and the sarcomere will remain contracted until the load stops and normal circulation is restored.

I believe this is a survival mechanism that enables an animal to hang on, even when the load would otherwise be overwhelming.

It also can cause muscle stiffness when circulation through a muscle is impaired, a very common condition as people age.

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The big difference between holding up a book in your hand (by holding it in the palm) and holding up a book by laying it on a table is that first equilibrium position is a dynamical one, while the book on the table is in static equilibrium.

I'll explain it qualitatively. You can compare the situation in which you hold up a book with the situation in which a book is held up by constantly bombarding it from below with particles, say marbles. In the extreme case of bombarding the book with only one marble at a time, the book falls a little, the marble hits it from below and thereby sends it back up again. The marble loses energy in the process, wich is given to the book (assuming an elastic collision). The book falls back again, and the next marble hits the book, sending it back up again, etc. You can use big marbles, little marbles, give them different velocities, and vary the amount of time between which the marbles hit the book. The best combination of these will hold the book in the best quasi-stable position. Even better would be to use many marbles, hitting the book at different places.

So each time a marble hits the book it loses some of its energy, which is given to the constantly falling book, and which makes it look like the book is in equilibrium. That is, a dynamical equilibrium.

Now, where is the connection with the muscles keeping up the book? I think it's easy to see, though I don't have too much understanding of the muscles workings. Alls muscle cells can be compared with the marbles and give the book constantly an upward change in motion during its fall. They relax, go tense, relax, go tense, etc. The fall and upward change are too small to notice, so the book looks in a steady state. That is, a dynamical steady state. Of course, there is no friction in the case of the marbles, who get their energy from "little canons".

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    $\begingroup$ That is a really nice example. It reminds me of a toy where ball is placed at the end of a pipe, you blow from the other end and the ball floats in air as long as you keep blowing consistently. $\endgroup$ – SMUsamaShah Jun 17 '17 at 20:53
  • $\begingroup$ Haha, that's a good one! I did play games with this toy with my friends in my (much) younger days. The one who could hold the ball as long as possible in the air won and had to "fight" the next opponent. Maybe that's where I (unconsciously) got the idea from! $\endgroup$ – descheleschilder Jun 17 '17 at 21:00
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helicopter

An analogy is the floating helicopter.

When it is static, no work is done to the helicopter.

But it is consuming the gas.

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  • $\begingroup$ This is not an answer - it's a good analogy, but you haven't explained WHY it needs to consume gas despite no work being done. $\endgroup$ – Benubird Aug 29 '18 at 14:19
  • $\begingroup$ Yes. In order to maintain a upward force against the gravity, the helicopter has to do work to the air. The air gets accelerated, and the reaction force is pushing the helicopter. $\endgroup$ – wwwjjj Aug 29 '18 at 20:39
  • $\begingroup$ If there was support below helicopter it wouldn't need to exert any gas to stay in position. Does it mean that the support was exerting energy upwards to keep the helicopter there? if so, where is that energy coming from? $\endgroup$ – SMUsamaShah Aug 30 '18 at 8:05
  • $\begingroup$ @LifeH2O The support from the ground is not exerting energy. But to maintain a support from the air, you need to give energy to the air. $\endgroup$ – wwwjjj Aug 31 '18 at 0:53
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Consider an analogy,

We get tired after STANDING for some time,without doing any work*. The reason behind this is same as the reason of why we dont do any work holding any object above our heads, but this case is easier to comprehend,

when we stand we r actually resisting the tendency of falling on the ground,muscles are holding on to the structure of our body so that we dont collapse on the ground like some non living thing,

these muscles have fibers which have have streached themselves ,which requires energy,

Similarly when we hold something above our head we r doing the same thing, resisting that collapsing tendency , which causes elongment in the muscles which requires energy.

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When a physicist talks about work, they are using the word in the technical sense of the equation you quote. To a biologist, though, work might be defined as energy expended to carry out a task. In your example, your arm will not naturally stay in the position described. Your body (mostly your muscles) must expend energy to hold your arm (and the book) in a set position, unsupported by anything but your own physiology.

So, by the biologist's definition, your muscles are doing work to hold up the book and your arm (muscle fibers are contracting and relaxing based on a host of chemical processes at the cellular level). But by the physicist's technical definition, no work is being done.

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    $\begingroup$ That doesn't really address the question of where the energy is going. $\endgroup$ – Mark Eichenlaub Dec 16 '10 at 13:21
  • $\begingroup$ The energy is going where most energy goes: into heat and entropy. $\endgroup$ – Jeremy Dec 16 '10 at 13:42
  • $\begingroup$ So the physicist's definition of work treats motion too small to be seen with the naked eye as if it were zero? I don't think that is right. $\endgroup$ – Trevor Wilson Jun 30 '14 at 6:37
  • $\begingroup$ Only if the physicist is analyzing a mechanical system. My heater is doing electrical work it has very little to do with the current definition. $\endgroup$ – G. Bergeron Dec 8 '16 at 10:32
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$F=ma$ means that every force is applied to a mass and produces an acceleration. Okay. Acceleration is $a=\frac{\Delta v}{\Delta t}$. If you put this $\Delta v$ into ${\frac{1}{2}m(\Delta v)^2}$ you discover the energy which have been necessary to let that mass accelerate. Since energy is neither created nor destroyed, it is the energy burnt by the one who applied the force! His/her/its potential energy (e.g. from food) has become kinetic energy of the accelerated body. Now, what about holding up 5 kg with your arm? No energy? Of course you spend energy. It is the same as above: you apply a force, equal and opposite to the gravitational force, so the object doesn't fall and doesn't rise and if you apply a force, for the reason above, you spend energy. Now one could object that there is no acceleration in this case. If no acceleration (opposite to the gravitational acceleration $g$) existed, the object would fall! We have two opposite accelerations (since two opposite forces) at stake ($\mathbf{F}=-\mathbf{F_g} \Rightarrow \mathbf{a}=\mathbf{-g}$). Which cancel. But if they cancel they both exist. So yes, you spend energy for holding the object up: to let this counter-acceleration exist. So you need energy to hold up a mass but no work is done if the object is at rest on your hand, since its kinetic energy is NOT varying. If you stop with your hand a falling body you cause a negative $\Delta E_k$ (you do negative work on it) but once it is stopped no more work, your energy is simply to cancel $F_g$ and keeping the body at rest.

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  • $\begingroup$ The applied force being equal and opposite to the gravitational force is not explained by Newton's 3rd Law. Keep in mind that in Newton's 3rd law pairs the two forces involved are always the same kind of force e.g. both gravitational forces or both frictional forces etc. $\endgroup$ – M. Enns Jan 5 at 0:52
  • $\begingroup$ @Enns: right, corrected (oversight), I was/am affirming that two opposite and equal forces are cancelled. $\endgroup$ – mfc Jan 6 at 1:35
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Energy is being expended maintaining it in position. Earth's gravity is applying a force downwards, the book is being accelerated down gravitation force.

A force is being applied to the hand and arm which must be resisted and thus energy expended.

The arm and book are not a closed system.

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    $\begingroup$ Err, no, the book isn't being accelerated down. $\endgroup$ – Billy ONeal Dec 16 '10 at 14:51
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    $\begingroup$ It's trying to accelerate, but it's not succeeding. This acceleration cannot account for the distance in the equation for $W$. $\endgroup$ – Malabarba Dec 16 '10 at 16:22
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    $\begingroup$ @Nim: so what? The table is also providing force to oppose gravity if there is a book upon it. Do you think it is doing work? $\endgroup$ – Marek Dec 16 '10 at 17:17
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    $\begingroup$ @Nim: I wasn't asking whether it will collapse (which is completely irrelevant because it would collapse sooner of later anyway, whether the book is on it or not). I asked whether it's doing work. And if so, explain where is the work being done (i.e. what part of table is exerting force over some distance). And note that if you'll again try to talk about something completely irrelevant I won't reply anymore. $\endgroup$ – Marek Dec 16 '10 at 22:19
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    $\begingroup$ This is an example of an "obvious" answer that is completely wrong. Obvious because everyone knows how hard it is to hold things up for a long time. Completely wrong because it misunderstands the physical definitions of "work" and "energy" and conflates them with "force". Read mbq's answer for a pop-sci level explanation of how bio-physics connects the everyday understanding of "holding stuff up is hard" with the physics understanding of "work". $\endgroup$ – dmckee Jan 29 '11 at 17:56

protected by Qmechanic Nov 19 '13 at 7:19

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