# Why is small work done always taken as $dW=F \cdot dx$ and not $dW=x \cdot dF$?

I was reading the first law of thermodynamics when it struck me. We haven't been taught differentiation but still, we find it in our chemistry books. Why is small work done always taken as $dW=F \cdot dx$ and not $dW=x \cdot dF$?

Very nice question! You can see this from the second Newton's law:

$$m\ddot{\mathbf{x}} = \mathbf{F}(\mathbf{x})$$

Now I would like to integrate this equation of motion with respect to time, to arrive at the energy conservation. To do so I multiply both sides with $\dot{\mathbf{x}}$:

$$m\ddot{\mathbf{x}}\cdot\dot{\mathbf{x}} =\mathbf{F}(\mathbf{x})\cdot\dot{\mathbf{x}}$$

and finally integrate:

$$m\int dt \ddot{\mathbf{x}}\cdot\dot{\mathbf{x}} =\int dt\mathbf{F}(\mathbf{x})\cdot\dot{\mathbf{x}}$$

The l.h.s. gives me the kinetic energy. The r.h.s. gives me exactly the integral in question:

$$\frac{1}{2}m\dot{\mathbf{x}}^2 = \int d\mathbf{x}\cdot\mathbf{F}(\mathbf x)$$

So the work done by the force is the kinetic energy of the particle (up to an integration constant representing its total energy).

Definition, A force is said to do work if, when acting, there is a displacement of the point of application in the direction of the force.

In layman language, to do work you need displacement, not just force.

In the equation, $dW=x.dF$, we are considering a change in force at a constant position from a reference point (origin). As per our definition, no work is being done because there is no displacement. To understand it better, assume there is a heavy block and you are applying a variable force to it.

No matter how hard you push, you won't be able to give it some speed. The work-energy theorem says that if a force does some effective work, there is a change in kinetic energy of the body. But in our case, there is no change in kinetic energy which implies no work is done by you. This is a pretty good way to get a hang of what work is.

On the other hand, in the equation $dW=F.dx$, we are considering an infinitesimal displacement for a constant force. Work is being done here as we have a force and a displacement. If we have a variable force, we will have to break our procedure of calculating work into infinitesimal displacements for which force can be assumed constant.

$$W=\int\vec{F}.\vec{dx}=\int\vec{|F|}\vec{|dx|}\cos\theta$$

• A good answer, but I think I would reverse the emphasis and place the energy argument first, from which follows the definition that makes it work. Jan 30, 2018 at 20:01

1. On one hand, work (within Newtonian mechanics) $$\mathrm{d}W=\vec{F}\cdot \mathrm{d}\vec{r}$$ is a scalar quantity, meaning that it is independent of coordinate system.

2. On the other hand, the quantity $\vec{r}\cdot \mathrm{d}\vec{F}$ depends on coordinate system. E.g. if we choose the $\vec{r}=\vec{0}$ to be the origin, the quantity vanishes.

Even if you are unswayed by physical arguments, you should think twice about introducing non-geometric quantities.

• Please please please! $\vec{F}d\vec{r}$ is scalar so is $\vec{r}d\vec{F}$! If $\vec{r}=0$ then the quantity vanish so does work vanishes if $\vec{F}=0$! Jan 31, 2018 at 13:21
• @physicopath: No. For starters, $\vec{r}$ does not transform as a $(1,0)$ tensor under coordinate transformations (even if we restrict to affine coordinate transformations). In contrast, $\vec{F}$ transforms as a $(1,0)$ tensor under general coordinate transformations. Jan 31, 2018 at 14:02
• @Qmechanic when I look at your answer the only thing I see is a snake eating its own tail. In addition, it surprises me that a professional like you answering such a simple question with such a simple answer, in this way. Jan 31, 2018 at 14:19
• @physicopath: In terms of transformational properties $\vec r$ is not a vector (to be more precise, it is not an element of the tangent bundle), so the scalar product involving it is not a scalar and thus not invariant under coordinate transformations. Jan 31, 2018 at 16:04

Because work $W$ is a force $F$ causing a change in position $\Delta x$.

Not just a force $F$ causing a position $x$. Or a change in a force $\Delta F$ causing a position $x$. Neither makes much sense. We are talking about a change in position - that is how work is defined.

And such a change $\Delta x$ is simply symbolized $dx$ when it is very, very (infinitely) tiny.

• I just do not understand why people, including @Qmechanic , trying to answer this question in these rather unintuitive ways while the answer is so simply as you put "Because work W is a force F causing a change in position Δx." not the opposite. Jan 31, 2018 at 13:09
• @physicopath Allthough I appreciate the gesture, keep in mind that the method of answering depends on how the answerer interprets the issue, how the answerer interprets the level of the OP and which approach the answerer from experience finds appropriate and most gaining. People choose differently; let's avoid calling out users by name. Feb 1, 2018 at 15:28
• "We haven't been taught differentiation but still, we find it in our chemistry books." this sentence is enough to interpret the issue. Feb 1, 2018 at 15:36

The two give very different physical results. Consider a force of $1~\rm N$ applied over a distance of $1~\rm m$. The work is correctly computed as:

$$W\int^{1~\rm m}_{0~\rm m} \vec F\cdot d\vec x=\vec F\cdot\int^{1~\rm m}_{0~\rm m}d\vec x=1~\rm J$$

Trying to apply the other formula doesn't give anything sensible. The force doesn't change, so evidently $d\vec F=0$:

$$W=\int^{1~\rm N}_{1~\rm N}\vec x\cdot d\vec F=0?$$

This would imply a constant force applied over any distance always gives zero work. Clearly, this is nonsense.

• Why can't you have a varying force over a constant distance? Jan 28, 2018 at 7:59
• @Allure Of course you can- this answer was just one simple example showing that it gives wrong results. The second formula doesn't give the correct results for that example, either: physically that should give you zero work (as no energy is imparted if the thing doesn't move). The first formula correctly gives zero work, while the second does not.
– Chris
Jan 28, 2018 at 10:34
• Why you can't have $$W=\int^{1~\rm N}_{0~\rm N}\vec x\cdot d\vec F$$O.o ? Jan 28, 2018 at 11:47
• @user45914123 That would indicate that the force changed from 0 to $1~\rm N$ over time. It isn't the right expression for a constant force.
– Chris
Jan 28, 2018 at 20:02
• @Chris Why it would represent a change with time, shouldn't it represent a change with displacement ? Why you need constant force ? Why can't a force vary with displacement sort of like electric field by a point charge - father you go lesser the force you experience ? Jan 28, 2018 at 20:10

$F(x,t)$ can always be expressed as a function of position (and time); however, it is generally wrong to write position as a function of force (as well as, $x(F)$ will fail to describe how nature works.)

To see why: considering an object being acted by a constant force, such a function is always multi-valued, it is so ill-defined that the input is always same, while the output is always different. Hence this $x(F)$ fails to predict the motion of the object at any given moment.

Now, let's see how it is wrong with physics: suppose we manage to find the area below the $\int x(F,t)dF$. Hence we know the "work done". However, the area below an object under constant force is zero, as it is a vertical straight line, but it is plainly wrong! The object is accelerating; therefore, something must keep inputing energy to the object! The work done can't be zero!

All in all, mathematically, "$x\,dF$" is ill-defined. Physically, it describes the physics so wrong.

I spent some time puzzling over this because I didn't find any of the answers satisfactory. I thought for a while that $dW = dF \cdot S$ is also valid, but is rarely seen physically. However, I eventually convinced myself that indeed, $dW = F \cdot dS$, and cannot be vice versa.

The key insight is that work only happens if there is a displacement. Holding a heavy object at a constant height is not doing work (in physics). Displacement is, by definition, $dS$. If $dS$ is zero, then there cannot be work. Only $dW = F \cdot dS$ satisfies this requirement.

But what if we have a varying force acting over a constant displacement? Let's say I apply the force $F = \mathrm{sin}(x)$ over a displacement of one meter. This superficially meets the requirements: a varying force, and a constant displacement. But when we look at the obvious equation for the resulting work done

$W = \int \mathrm{sin}(x) \times 1 \mathrm{m}$

It becomes clear that even then, the formula used is actually $dW = F \cdot dS$! The equation has only one variable, which means there can only be one integration variable - $dx$ (which is equal to $dS$ in 1D). The "constant displacement" isn't constant at all. In fact that one meter is equal to $dS$

tl; dr: $dW = F \cdot dS$ is the only equation that makes physical sense. I don't know if this is the answer you're looking for, but it's the one that answered the question for me.

The answer is very simple, (small, useful) Work is defined as dW = F.dX.
There is another type of ("not useful") Work that is given by dW = X.dF.
An example of this type, is a person lifting a given weight by applying a force less than mg. As the force changes from 0 to almost mg, no (useful) work is done (the weight does not move).

• No. Work is always $dW=\vec F\cdot d\vec x$. It's not even really a definition, but a direct consequence of Newton's second law.
– Chris
Feb 3, 2018 at 7:44