# The Calculation of work in different frames

Say there is a man running on a road. Friction is applying a force hence he is accelerating, but the friction is static hence does no work. We know that the change in kinetic energy is due to internal work done by his muscles.

Now say that the same man is running on a wooden plank kept on a frictionless surface. As the man moves ahead the plank goes behind. At the point of contact, the friction applies force in the forward direction accelerating the man ahead, but since the plank itself is moving behind, the point of contact is going behind. Hence, work done from ground frame is negative. But if I change my frame to the plank then work is zero again. Thus work is frame dependent.

Is my conclusion correct? And I always think if the calculation for work is frame dependent than when we theorize physics we might be missing out on different types of work which may lead to concepts like dark energy. How to physicists think about this.

Thus work is frame dependent. Is my conclusion correct?

Yes. Work is frame dependent. This can be easily seen by noting that distance is frame dependent and since $$W=F\cdot d$$ then work is also frame variant.

Note that the plank frame that you describe is non inertial. So energy is not trivial and is actually not even conserved in that frame. However, despite that complication your conclusion was correct.

if the calculation for work is frame dependent than when we theorize physics we might be missing out on different types of work which may lead to concepts like dark energy

No, there is no link between the two. Dark energy is a specific term in the current best model of the universe.

From the ground reference frame, there is no net work done. Friction did displace you by a distance of say $$d$$. However, the same friction also displaced the wooden board by a distance $$d'$$. These two perfectly cancel and this can be realised by noticing that the center of mass remains fixed throughout this process.

However, with respect to the wooden board or the man, they have indeed done work. This is because both are non-inertial frames where Newton's laws do not hold. There is no apparent equal and opposite force applied back and it seems that work was done out of thin air. For this reason, a pseudo-force is added to make sense out of non-inertial frames.

The concept of dark energy has nothing to do with the frame of reference and was theorised through a series of experiments to explain the rate of expansion of the universe.

• You're assuming the person and board have the same mass? Feb 8, 2020 at 23:20
• In the ground frame kinetic energy is created so total work done cannot be zero, can it? Feb 9, 2020 at 2:21
• @AaronStevens No, why would I assume that.
– Sam
Feb 9, 2020 at 5:13
• I think you know why, hence your edit :) Although what is cancelling out in the first paragraph, exactly? Feb 9, 2020 at 5:17