# Work done by non-conservative force like friction

I have two questions:

Q1 Why is work done by friction path dependent?

Let us say that there are two points A and B that are vertically at a distance 'd' apart from each other. If I take an object and move it from point a to point b such that the path length covered is 'd' and in another case if I take that object and move it from a to b such that the path length covered is 2d then how is work done in those two cases different?

Q2 why work done by non conservative forces is equal to force * distance instead of force * displacement?

Q1 Why is work done by friction path dependent?

Q2 Why work done by non-conservative forces is equal to force * distance instead of force x displacement?*

Since friction forces are non-conservative, this will answer both questions.

Friction work is non-conservative because it is dissipative (generates heat). The friction work is the friction force (assume kinetic friction) times the distance covered in the path from A to B. The greater the distance covered in the path between A and B, the greater the friction work and the more the heat generated. What’s more, if the mass is returned from B to A, the total work done will be the sum of A to B and B to A, and will depend on the total distance traveled. If the mass begins at rest and ends at rest, all of the work done generates heat.

Contrast this to the work done in a gravitational field. Gravitational force is conservative. Let B be at a greater height than A near the surface of the earth. The only work that increases the potential energy is the force times the positive vertical displacement between A and B. Any horizontal displacement involved in the path between A and B does not change the potential energy. If we now return the mass from B to A, only the total negative displacement will result in a decrease in potential energy and it will be equal in magnitude to the increase going from A to B. If the mass begins and ends at rest, the net work done is Zero.

Hope this helps.

• thank you for the answer.. so forces that when applied on an object dissipate heat can be classified as a non conservative force , right? Nov 29, 2018 at 17:13
• also ,you stated that the work done by friction should be positive but shouldn't the work done by friction be negative because friction opposes the direction of motion? Nov 29, 2018 at 17:23
• Regarding your first comment, you are correct. Regarding the second, you are also correct. When I said positive I was thinking in terms of the force delivering energy to the contacting surfaces. I've eliminated the reference to positive work in the answer. Nov 29, 2018 at 18:49

If you were to take a block of sandpaper and rub it around in a full circle on a table, the block would afterwards be in the same position as if it had not moved at all. Hence, its displacement is zero. But it is clear that work is done when the sandpaper is rubbed in a full circle, whereas no work has been done if the block had not moved at all. This shows that the work is path-dependent; since the displacement is zero in this case, it also shows that the work cannot depend on only the displacement.