# Why are work and energy considered different in physics when the units are the same?

There is a question that explains work and energy on stack exchange but I did not see this aspect of my problem. Please just point me to my error and to the correct answer that I missed.

What I am asking is this: Why in physics when the units are the same that does not necessarily mean you have the same thing.? Let me explain. Please let me use m for meter, sec= second , and kg = kilogram as the units for brevity sake.

The units for work are kg * m/sec^2 * m. The units for kinetic energy are kg* (m/sec)^2. They look that same to me. I need them to be the same so I can figure out the principle of least action. Comments are welcome.

• height and width are all in meters. Thrust and weight are both in Newtons. Their differences are operational but they exist in the same dimensions – lurscher Sep 29 '17 at 15:23
• If you take this argument to the extreme, in the system $\hbar = c = G = 1$ literally nothing has units. So why isn't absolutely every quantity considered the same? – knzhou Sep 29 '17 at 15:25
• Some external agent can do work, but it can't do energy. – Mitchell Sep 29 '17 at 16:51
• obligatory xkcd – user118047 Sep 29 '17 at 17:51
• Work is a process function (just like heat). Internal energy, kinetic energy and potential energy are all state functions just like temperature, pressure and volume. The former is defined for a process, the latter for a body. It means that they can never be the same. – Eric Duminil Sep 30 '17 at 16:35

One definition of work is "a change in energy." Any change in a physical quantity must have the same units as that quantity.

Different kinds of work are associated with different kinds of energy: conservative work is associated with potential energy, non-conservative work with mechanical energy, and total work with kinetic energy. In fact, that's one way to see the oft-quoted Law of Conservation of Energy:

$$W_{total}=W_{non-conservative}+W_{conservative}\\ \Delta KE=\Delta E - \Delta PE \\ \therefore \Delta E=\Delta KE + \Delta PE$$

So just like impulse (which is a change in momentum) has the same units as momentum, work has the same units as energy. Any change in a physical quantity must have the same units as that quantity. A change in velocity has units of velocity, etc.

A more difficult question might be why torque has the same units as energy. This is more subtle, but the key concept is this: units are not the only thing that determines a quantity's interpretation. Context matters too. Energy and torque may have the same units, but they are very different things and would never be confused for one another because they appear in very different contexts.

One cannot blindly look at the units of a quantity and know what is being discussed. A dimensionful quantity might be meaningless or meaningful depending on the context, and it's meaning can change with that context. Action times speed divided by length has the same units as energy but without any meaningful interpretation (as far as I'm aware).

• FWIW, it could be argued that the units of torque are $\mathrm{\frac{J}{rad}}$ (i.e. Joules per radian), but that just gives a name to the pseudo-unit that is the radian – unitless in some sense, but a unit in the sense that it signals that your talking about unitless angles. – Sean E. Lake Sep 29 '17 at 17:31
• That is a worth-while observation. Analogously, we could describe the units of force as J/m. At this point, a natural question is why radians are unitless. The answer is because they are defined as the ratio between two quantities with dimension of length: arclength to radius (hence the name). When we use torque, we are basically multiplying the force by the radius so that when we later multiply by radians we get force times arclength (which is work). In a sense, it is just useful shorthand. – Geoffrey Sep 29 '17 at 19:08
• Similar to @Sean's comment you can distinguish energy and torque by their tensor character: energy is a scalar and torque is a pseudo-vector. – dmckee Sep 29 '17 at 19:09
• "A change in velocity has units of velocity, etc." Note that while acceleration describes a change in velocity, acceleration as a quantity describes the rate of change of velocity, whereas this post refers to the total change. (Just a minor point where someone might get tripped up.) – jpmc26 Sep 29 '17 at 19:37
• The torque is just the energy gained / spent to rotate the lever by one radian – John Dvorak Sep 30 '17 at 14:03

Perhaps a better analogy than height and width can be found in terms of money. Both the balance in your bank account and amount you pay for, say, your electricity bill are denominated in the same units (dollars where I live), but they represent separate concepts. One is measure of what is stored and the other is a measure of what is transferred.

In most uses 'energy' means an amount available in some system (like the balance in an account) while work and heat represent transfers (like payments).

The analogy can be pushed too far as we often don't care about absolute level of energy and treat negative balances in exactly the same way as positive one while your bank may take a dim view of your maintaining a negative balance.

• Excellent answer. I'll keep this analogy in mind next time I meet an engineer not knowing the difference between state and process functions. Sadly, it happens relatively often. – Eric Duminil Sep 30 '17 at 16:48
• In my experience a substantial fraction of them have a vague, implicit understanding of the difference but can't articulate it (I know I was in that boat for more than a year in my undergraduate education). – dmckee Sep 30 '17 at 17:43
• Also, money is not a conserved unit... For better or worse. – Vendetta Oct 6 '17 at 16:56
• @Vendetta Well, at the interest rates you can get from a saving account (or even a CD) these days money can be treated as conserved to a good approximation... – dmckee Oct 6 '17 at 17:28
• @dmckee Locally, and for a short period of time, sure. – Vendetta Oct 6 '17 at 17:31

Work and energy are in fact different things, but they are closely related. So closely that they share the same units. To understand the difference between them and why this difference does not imply new units, let me borrow a story told by Feynman in his Lectures on Physics and improved by Van Ness in his Understanding Thermodynamics.

Let us imagine a boy living in a house with his mother and 37 indestructible small cubes. The house has two windows labeled by $W$ and $Q$. Every day the mother counts the cubes and some day she finds only 35. The boy does not say where are the two missing cubes but the mother notice that they can be inside a box which (by some reason) she cannot open. She weights the box, wait until another day when she counts 37 cubes, weights the box again, takes the difference between the readings and divide it by the weight of one cube. She finds out that the two cubes missing the other day were inside the box. Another day she counts only 30 cubes. She weights the box again, does the math and notice there are still 3 cubes missing. She realizes that those missing cubes can be on the bathtub. She cannot either see or check with is hands because the water is dirty. She measure the level that day and in another day when there is no cube missing, does the math and can now account for blocks in the bathtub. She has a formula able to account for blocks eventually hidden in the box and in the bathtub: $$37=\mathrm{visible\ cubes}+\frac{\mathrm{weight\ box}-\mathrm{weight\ empty\ box}}{\mathrm{weight\ one\ cube}}+\frac{\mathrm{level\ bathtub}-\mathrm{level\ bathtube\ with\ no\ cube}}{\mathrm{change\ of\ level\ due\ to\ one\ cube}}.$$

One day however she checks the box and the bathtub but she cannot find 37 cubes. Some day she finds even 40 cubes! The only conclusion is that there are cubes being thrown in and out the windows. She came up with another formula: $$\mathrm{number\ cubes}-37=Q+W$$ where $Q$ and $W$ denotes the number of cubes that crosses the windows $Q$ and $W$. If the cube comes in, the number is positive, if the cube is thrown out, the number is negative.

So, the first equation is a conservation law and the 37 cubes metaphorically represents energy. The three terms on the right means different ways energy (cubes) can show up. It could represent, for instance, kinetic energy, rest mass energy, the different potential energies. All with the same units. The second equation represents a way of accounting for energy (cubes) entering and living the system (the house) via heat (window Q) or work (window W). Again, all the terms in that equation has the same units. That formula actually represents the first law of thermodynamics and $W$ stands for work. As you can see, work is the term given for energy that enters or leaves the system in a given way (window W), namely, through ordered motion. On the other hand, heat denotes the energy entering or leaving the system disorderly (window Q). Thus, by its very own construction, work has to be the same units as the energy of the system, even though they are different things.

• This seems like a re-working of the explanation of the conservation of energy in Feynman's Lectures on Physics with just a few details changed. – sammy gerbil Sep 30 '17 at 10:51
• @sammygerbil It is, as I said in the first paragraph! – Diracology Sep 30 '17 at 13:23
• Sorry, I missed that. – sammy gerbil Sep 30 '17 at 13:25

Energy, as a concept, relies on the concept of systems, or drawing good boxes. When we draw a box, around a planet or an engine, we're basically saying this stuff in the box is our system. When Energy crosses the border of that box (into or out of the system), that's one kind of Work. When the box changes shapes (for example when the gas in a cylinder expands doing $PdV$ work on the piston), that also represents Work. In classical mechanics, we often ignore a lot of that and just say that there's a conservative field (gravity) that can do Work on the (much smaller) object. In other words, Work is a type of Energy.

Regarding your question about units, if two things have the same units, they may not necessarily be added. For example, Torque has units of $F\cdot l$ (force $\cdot$ length) which is mass $\cdot$ length$^2$ per time$^2$. Energy has the exact same units, but we may not add a torque to an energy and get a meaningful result. In other words, having the same units is a necessary, but not sufficient requirement to add two physical quantities. As far as how to know if you can add two quantities, it's contextual. As disheartening as that may seem, it's not so bad - you'd never add the nuclear binding energy of tea leaves to the kinetic energy of a baseball (are you envisioning that experiment now?) even though they have identical units.

• Your clue got me thinking...and I am Ok now....you said "having the same units is a necessary , but not sufficient requirement to add two physical quantities" . I had to think of an example where the units are the same but the objects are different. BUT if you have an equation x = y then the units had better be the same and that is the necessary condition you refer to. – Sedumjoy Sep 30 '17 at 2:19

Work is a kind of energy. Kinetic energy is another type of energy. One is a type of energy transfer, the other is motion energy.

Isn't this the same as asking why electric force and magnetic force aren't the same? They have the same units and work in the same way, but there origins are different.

• you make a great point....thank you ...but if the units of the electric and magnetic force are the same how do ya tell apart? – Sedumjoy Sep 29 '17 at 15:39
• I mean if you didn't know where the energy came from you still can tell with measuring instruments can't you ? with electricity and magnetism I mean..so you say the units are the same ...but then how can he measuring device possibly tell the diff? help me out...this is a separate issue , I think you helped me out on the KE and Work problem I had – Sedumjoy Sep 29 '17 at 15:42
• @Sedumjoy If you use a ruler, how can you tell if you're measuring a width or a height, if they have the same units? – knzhou Sep 29 '17 at 16:10
• @Sedumjoy It's worth clarifying that he is saying that the electric force and the magnetic force have the same units. The electric and magnetic fields have different units. A device that measures forces might be agnostic to the origin of the force, but another device could certainly tell the difference between a magnetic and an electric field. – Geoffrey Sep 29 '17 at 16:38
• I don't think I like the statement "Work is a kind of energy", exactly because it's not a form of energy. It's a number that specifies how much energy has been transferred from one object to another via the action of macroscopic forces between the two bodies, but it's not a type of energy in and of itself. It's a fundamentally different animal: you can assign an energy to a system, but you cannot assign a work to a system. – march Sep 29 '17 at 17:57

Why in physics when the units are the same that does not necessarily mean you have the same thing.

Consider these two different things:

• The amount of floor space in your house
• The fuel economy of your car

Both are measured in square-meters.

References

Why can fuel economy be measured in square meters?

If your velocity changes from 5 m/s to 8 m/s, you say your velocity has changed by 3 m/s (assuming same vector direction) and your new velocity is 8 m/s. This seems like a very obvious statement; 3m/s represents change and 8 m/s the measure. In essence, a change in a vector or scalar quantity will have the same units as the quantity itself.

Work is nothing but change in energy and hence has the same units as energy itself.

It is a short answer but this is it!

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