# When is work done by static friction force positive and when is it negative?

So we say that a work done by a force is positive when both the force as well as the displacement of the body are at acute angles with each other. With this logic, consider a man standing on a plank which is placed on a smooth horizontal surface. Now the man starts running over the plank and the questions asks us whether the work done by friction on the man w.r.t ground frame is positive or negative. Applying Conservation of momentum, we know that from ground frame, the displacement of man will be towards right and also the friction force on the man is towards right, so answer should be positive work done, BUT, the answers says the following: At the point of contact, man's foot is moving backwards and friction force acts forwards, so work done is negative! How? i don't understand this solution, please help.

• How is the mans foot moving backwards? Only when it is off the ground.
– Max
Dec 3, 2020 at 3:15
• how do we move forward, we press the ground backwards and so our foot moves backwards thus pushing us forward.
– Vega
Dec 3, 2020 at 3:18
• Dec 3, 2020 at 3:38

The reason for your confusion is very simple: you have a misunderstanding of exactly what "displacement" means in the definition of work.

When you're calculating work done on an object (for example, a person) by a certain force, the displacement you need is NOT the displacement of the whole object! Instead, you need the displacement of the exact point/part of the object to which the force is applied.

That's it, that's just how work is defined. So it doesn't really matter at all where the whole person is moving, the displacement you care about is (by definition of work) the displacement of the exact part of the person to which the friction is applied, i.e. the foot.

Addendum: this is less important, but it seems that some people who commented on this question misunderstood the situation (which wasn't described very carefully by the OP). The person is trying to walk to the right along the plank, but the plank itself is on a slippery surface, so it's slipping to the left. So his foot is stationary with respect to the plank, but moving to the left with respect to the ground.

• thanks a lot, this got my basics cleared!
– Vega
Dec 4, 2020 at 2:33

In the frame of reference where the plank is stationary, the man is accelerating forward. In that frame, his foot is stationary, so static friction does no work. His leg pushes him forward. His leg does positive work.

In the frame where the man's center of mass is stationary, the plank and the world are accelerating backwards. This is a non-inertial frame, so it will be more complicated. There will be a fictitious force acting on the world, accelerating it backward. The force acts on the man, accelerating him backward. It is just enough to balance the force from the man's leg pushing forward. The total force on the man is $$0$$, which is why he stays still. Fictitious forces take a little getting used to.

In this accelerated frame, the man's foot is moving backward, trying to accelerate the plank and the world. The plank and world are moving backward. So the work done on the world is positive. The world is so massive that the man's foot doesn't change its motion noticably.

This might also help. Is work done in rolling friction?