# If application of force does not result in spatial movement, has work been done? [duplicate]

I'm trying to help my kids why their elementary-school physics. Their lesson today says that "work" is done only when a change in position is accomplished by the applied force. I have absolutely no background or training in physics whatsoever.

So my question is if two people were to stand facing each other with a book (or another object) between them, and they both applied equal force to opposite sides of the book (each person attempting to push the book toward the other person) would that be "work" in the physics sense?

Since each person is accomplishing a change in the book's movement (stopping it from moving) would that not be considered work?

I guess my question involves two forces instead of one. Maybe that makes a difference?

• "Since each person is accomplishing a change in the book's movement (stopping it from moving)" - Work is done only if something changes position - if the book didn't move, no work has been done. Apr 30, 2020 at 17:24
• @OfekGillon: However, according to this question about walking in a circle work has been done if one were to walk a closed path, returning to their point of departure. Two people pushing on a book are adding energy to the book - so would work not be done? Apr 30, 2020 at 17:33
• Possible duplicates: Why does holding something up cost energy while no work is being done? and links therein. Apr 30, 2020 at 17:56
• Don't confuse common sense and logic with scientific concepts. It makes sense that if I try to lift a car, if I spent sixty seconds doing so, I'd be very tired, and I would've done a lot of work, because I'd be tired, right? Physics is applying mathematical concepts to real-life phenomena, and so there need to be agreed-upon definitions, which is why physics books introduce concepts by defining them. Thus according to the physical definition of work, if no movement occurs, no work is done. You may have similar conceptual difficulties when you get to the topic of energy. May 1, 2020 at 21:55

No. No work would be done in this case, at least not at the macroscopic level. Work is the product of force and displacement in the direction of the force and in this case there is no displacement.

I disagree with you that each person is "accomplishing a change in the book's movement". The book wasn't moving initially, or at the end, or at any time in between. The situation is exactly the same as if two people had been trying to push the book through a solid wall.

Be careful when using the human body in questions where you are asking how much work has been done. If you actually try what you propose you will find that you will get tired. Your muscles are losing chemical potential energy but you are not doing work on the book. At the microscopic level your muscle fibers are contracting and slipping and contracting again so at that level work is being done.

I find it more instructive to think of replacing the people in examples like these with some sort of simple mechanical device. You could lean something up against the book, use a clamp or set up some other simple mechanical system which would continue to apply a force but would require no ongoing energy input, which makes it clearer that no work is being done.

• You can kick a soccer ball all around a soccer field and put it back where you found it. You may be sweaty but you've done no work. You've just played around. Now if you'd have moved the ball up into the stands... May 1, 2020 at 10:33
• @candied_orange Just because $\oint_Cds=0$, doesn't mean $\oint_C F\cdot ds=0$; this won't be the case for a nonconservative $F$, which is why e.g. the ball could get back to where it started with more kinetic energy than originally.
– J.G.
May 1, 2020 at 11:56
• Good point about the human body giving bad intuition for when energy is and isn't used. Magnetism always causes confusing for the same reason - "why doesn't it require energy for a magnet to remain stuck to the fridge door" for example. May 1, 2020 at 11:58
• This is the right answer. Another example could be a car that is attached to a tree with a rope (trying to tip it over). As long as the tree doesn't move no work is done on the tree. But meanwhile the machinery in the car is losing lots of energy so the insides of the car are doing work, while no useful work is done. May 1, 2020 at 11:58
• Great point about taking the person out of the scenario. It's pretty intuitive that if you can replace a person with an inanimate, non-moving object, the person's physical ability to exert themselves is irrelevant. May 1, 2020 at 15:19

Their lesson today says that "work" is done only when a change in position is accomplished by application of force.

The key word here is change. If an object is not moving, then no work is being done to it$$^*$$. Therefore in your scenario, no work is done on the book by any of the forces acting on it.

More formally, work done by a constant force on an object moving in the direction of the force is given by $$W=F\Delta x$$ where $$W$$ is the work done by the force $$F$$ over the distance $$\Delta x$$. If the force is at some angle relative to the direction the object is moving, but the force is still constant and the object is moving in one direction, then we pick out the component of the force that is along the direction the object is traveling: $$W=(F\cos\theta)\Delta x$$ which has a shorthand notation for vectors $$W=\mathbf F\cdot\Delta\mathbf x$$ If we have more complicated scenarios of forces that are not constant and objects whose directions are changing, we just "zoom in" onto the path until we have small segments where the force is constant and the object moves in one direction. We find the work $$\text dW$$ along this little part of the path $$\text d\mathbf x$$ as before then: $$\text dW=\mathbf F\cdot\text d\mathbf x$$ and then we just add all of those little works up. Welcome to calculus:$$W=\int\mathbf F\cdot\text d\mathbf x$$ The $$\int$$ symbol qualitatively means "add up all of these little values."

Going back to your book example, there is no path the book is moving on. $$\Delta x=0$$, so $$W=0$$.

Since each person is accomplishing a change in the book's movement (stopping it from moving) would that not be considered work?

No. First, "change in movement" is not the same thing as "change in position". Second, the book is remaining stationary, so nothing about it is changing. Work doesn't take into account the idea that "well if this other force was gone then the object would do this instead". You just focus on what is happening, not what could be happening due to some other scenario.

$$^*$$Technically we are interested in the point of application of the force. But for non-rotating rigid bodies the distance covered by the point of application is the same as the distance covered by the object, so we can often use these two interchangeably.

It's complicated.

The simple idea for simple situations is to think of it as force times distance. And that does work.

In thermodynamics, work performed by a system is energy transferred by the system to its surroundings, by a mechanism through which the system can spontaneously exert macroscopic forces on its surroundings, where those forces, and their external effects, can be measured. In the surroundings, through suitable passive linkages, the whole of the work done by such forces can lift a weight.

Wikipedia

So any energy transfer is work if it could possibly be converted into moving a mass.

Say you push electric current through a resistor. At first sight, all that moves is the electrons. But heat is produced. The heat is partly molecules vibrating faster, and partly it is radiation that could someday cause other molecules to move. All that movement and potential movement is work. Because energy has been transferred.

A very interesting question. Work has several notations or meanings in Physics. Mechanical work done by you and defined as : $$W = F\Delta l$$ would be likely zero, because book is not changing it's position, due to force applied.

However, there is also a thermodynamic work achieved by you, which is defined as : $$W = Q + \Delta U$$ where $$Q$$ is heat transferred from book to it's surroundings and $$\Delta U$$ is change in book internal energy (molecules kinetic energy, deformation energy, chemical energy, and etc.). So when you both apply a constant, but opposing forces, book experiences some sort of compression and various stresses, which should accumulate into internal energy, part of which can be released back as book emitted heat. You can check this out with temperature mapping devices.

Physicists use a narrower definition of the word "work" than normal people. We care about the amount of work done on an object and less often about the work done by something else. If work is done on an object, then that object has to change in some way, whether by changing speed, getting hotter, or changing shape.

In your example with two people pushing on a book, if the book never moves, then no work is done on it. The people will be tired after a while, and their muscles will have done work on themselves--as evidenced by the muscles getting hotter. But, this has more to do with the biology of muscles. So, energy was expended and calories from food were used up, but none of that work was done to the book.

Simpler example: imagine that same book sitting on a table. There are two forces acting on the book: gravity and the table's stiffness [1]. These forces are equal in magnitude and opposite in direction [2], so they add up to zero and the book doesn't move. If an object is not changing in some way, then there is no work being done to it.

One clue as to whether work is done is to see if the motion, or lack thereof, could continue forever without some sort of input like fuel or electricity. It takes work to accelerate an object in a straight line, since any vehicle needs fuel to do this, whether it's a car, a rocket, or a sprinter. In contrast, a book sitting on a table can remain there indefinitely, so the table and gravity are not doing work.

In a comment, you asked whether walking in a circle constitutes work. It is natural to think this, since the force causes a change in motion. Plus, being a passenger in a car making violent turns subjects one to a lot of unpleasant interactions with the seat belt and window. However, consider the planets going around the Sun. Earth has been going around the Sun for over 4 billion years and the orbital path it follows does not seem to be changing appreciably. The Sun surely burns through a lot of fuel, but it does so to release heat and light, not to affect the motion of the Earth. In fact, if the Sun were replaced with a black hole of equal mass, the Earth's orbit would be the same. This is evidence that nothing is doing work on the Earth to keep it in orbit. This is why work is defined as the movement of an object multiplied by the component of force on that object in the same direction as the motion. The gravitational force from the Sun is nearly always perpendicular to the velocity of the Earth, so no work is done [3].

[1] Technically called the "normal" force because the force is normal--i.e., at 90 degrees--to the table's surface.

[2] This is not an example of Newton's Third Law.

[3] If a planet orbits a star in a perfect circle, then no work is done at any point in the orbit. However, most orbits are elliptical, so the planet speeds up as it gets closer to the star and slows down as it gets farther away. But, since the planet is traveling at the same speed when it returns to the same position in orbit, the total work done over a complete orbit is zero.

If the book is not moving under the action of the forces applied to it (your hand, your friend's hand, Earth's gravitational force, the reactive force from the table surface and friction between the book and the table surface) then the total force applied to the book is zero. If you neglect the book's deformation then no work can be done.

When we say that work has been done by a force then it means that the point where force has been applied has shifted from it's initial position.

In your case, the point where force has been applied doesn't shifts which means no work has been done.

Preventing the book from moving is not the same thing as accomplishing a change in the book's motion. The book was not moving in the first place in your scenario.

The definition of work done on an object by a force is equal to the component of force along the displacement of the object. There are a lot of concepts here and the proper definition uses integral calculus. First, Force is a vector, as is displacement, velocity, and acceleration. These quantities have direction as well as magnitude.

The short answer to your title question is NO. If the application of force does not result in movement then no work has been done. There are better examples than the two people fighting over a book (I'd prefer to imagine them both pulling out of interest in the contents). Take the example of an object sliding across a table, or being pushed or pulled across any horizontal surface. This object is being acted on my gravity which pulls it down, and a normal force of the table which balances gravity pushing it up (keeping it from falling through the table to the ground). Neither of these two forces does work on the object as it slides across the table since there is no component of force in the direction of motion, the displacement is horizontal while the forces are vertical. Things get more interesting if you consider a car on a roller coaster, or a mass attached to a string swinging like a pendulum. In the latter case the tension in the string does cause a change in motion of the mass at the end by causing it to change direction. However, it does not do any work on the mass since it acts perpendicular to the displacement at any instant (the displacement being tangent to the circular arc made by the mass and the tension being directed in to the center of the circle along the radius).

In the case of a person pulling an object across the ground using a rope that is at an angle with respect to the ground, some of that force does work and some does not. Assuming the object moves horizontally, the component of force along the horizontal will do work while the vertical component will not.

Work done on an object by a force can be thought of as energy given to that object and is observed as an in icrease in kinetic energy (energy of motion) by increasing its speed. It is possible for a force to do negative work on an object. An example of this is friction. Consider an object sliding across the floor (carpeted or covered in some otherwise rough material). This object will eventually slow down and come to rest. The frictional force exerted on the object by the floor acts in the opposite direction as the displacement which introduces a minus sign. This is taking energy away form the object.

Now consider many of these factors acting together. Think of a person dragging a large object across a rough horizontal terrain by a rope. Even though they are applying a force and doing work on the object friction is also doing negative work and taking energy away from the object. If the object is observed to move at constant speed its energy is not increasing or decreasing and the work done on the object by the person pulling is taken out by friction, hence there is work being done on the object by two agents and no net work being done on the object.

The last example serves to illustrate that we can talk about the work done by an agent on an object but we can also talk about the total work done on an object.

• Explain the down vote
– user196418
May 3, 2020 at 14:17