# What is the work done by individual forces on an object at rest?

An object at rest is being acted on by several forces which add up to zero.

For instance, a box is sitting on an inclined plane. It is not falling down the plan because the force of friction cancels the component force of gravity parallel to the plane, and it is not falling through the plane because the normal force exactly cancels the force of gravity perpendicular to the plane.

If we define work as force times distance, there's no work being done on the box, because the box isn't moving. But can we say that any of the individual forces did work because they prevented the box from moving a given distance?

Also, what if the entire system is moving at a uniform velocity relative to a given frame of reference? Let's say the system is moving towards the center of the earth at uniform velocity. After a certain amount of time, the system has technically moved some distance x, and the force acting in the direction of this movement, gravity, is not zero, so we should get a non-zero value for work. But relative to the system itself, no work was done! Is that really possible?

Whenever you're confused about forces and work, you can bring it back to energy. The real definition of work is the transfer of energy. If no energy is transferred, no work is done. Ever. Force times distance doesn't define work, it quantfies it. The only mechanism for the transfer of energy is force exerted through a distance.

But there doesn't have to be a nonzero net force. If you push a box across a floor against a friction force at constant velocity, both you and the friction force are doing work. Some people would say that equal amounts of positive and negative work were being done on the box. I don't like that. I prefer to say that chemical energy from you is being transferred to thermal energy in the box and the floor. That keeps it real.

In the case of an object or system traveling toward the center of the Earth at constant velocity, energy is being transferred by the force of gravity from the gravitational field to the person or thing applying the equal and opposite force.

• "The only mechanism for the transfer of energy is force exerted through a distance."... Or heat. Oct 21, 2016 at 16:08
• Even heating transfers energy via force exerted through distance, in the collisions between particles. Oct 21, 2016 at 16:12

If we define work as force times distance, there's no work being done on the box, because the box isn't moving. But can we say that any of the individual forces did work because they prevented the box from moving a given distance?

What you're doing there is redefining work. You've defined it to be force times distance then redefined it to be something else.

Also, what if the entire system is moving at a uniform velocity relative to a given frame of reference? Let's say the system is moving towards the center of the earth at uniform velocity. After a certain amount of time, the system has technically moved some distance x, and the force acting in the direction of this movement, gravity, is not zero, so we should get a non-zero value for work. But relative to the system itself, no work was done! Is that really possible?

If it's moving towards the centre of the earth at a uniform velocity, then it has two forces acting on it, one gravity and the other something else, like air resistance in the case of a parachuter falling at terminal velocity. Yes, work is being done in this case, the net force times the distance. The system is also losing its potential energy. Changing the frame of reference doesn't make any difference.

I like to think of work in the everyday use and intuition of the word:

• If you push in a wall, you do no work. It might be very tough and you feel you a huge effort, but I'm the end you didn't really do any difference. You did no useful work because you didn't manage to make it move.

• If you push a balloon, you do no (I mean, very little) work. You might feel like you did a big change and that you moved the balloon very far, but it wasn't very hard. You didn't really do any effort to move it. So noone would say that you did any significant work here.

Bottom line, stick with the formula that you mention yourself:

$$W=\vec F \times \vec x$$

You need both force and displacement. So it's very much okay to say that no work is being done by forces that do no change.

A table is not constantly doing work on a book that it holds up. As another answer says, work is a transfer of energy. And the table spends/transfers no energy while holding up the book.

If the kinetic energy of the object has not increased, then no work has been done on it. However, the forces can do work against each other.

For example, you could push the object at constant speed over a rough surface. The net force on the object is zero. But you have done work against the friction force, which is dissipated as thermal energy.