Newtonian Mechanics – Does Work Differ Between Inertial Frames?

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I have a question regarding the frame dependence of both kinetic energy and work.

  1. I was able to prove that if I have a system of particles with masses $m_i$ and momenta $p_i$, then the change in kinetic energy is invariant under Galilean transformation only if the total momentum is conserved.
  2. If momentum is not conserved, then the change in kinetic energy is not invariant, and hence (by the work-energy theorem), the work done is different between frames.
  3. My best thought till now is that work is $\text{power}\times\text{time}$ and power is $F \cdot v$. Since inertial frames do agree on $F$, but not $v$, power is frame-dependent. Hence, work is also frame-dependent, but this doesn't make sense to me! If I exerted some work to move things apart for example in one frame, everyone else should see me doing the same work, right? we both agree on the force $F$ and the path $dr$.
  4. Additional question, is, what about potential energy, is it frame-dependent? This should follow from the answer of "is work frame-dependent?", right?

Take a simple example of one ball of mass 1 kg pushed to go from 0 m/s to 1 m/s in one frame and from 9 m/s to 10 m/s in another frame. What is the work done on it? And what is the meaning of each number in that frame?

Best Answer

The conclusion: Work done is dependent on different frames. Let's look at a simple example. Say there is an object(1kg) that is initially at rest and we apply a constant force to move it for a certain distance to reach 2m/s in the rest frame(call it O). The KE change in frame O is 2J. Now consider a moving frame O' moving with a velocity=1m/s to the left with respect to frame O. In frame O', the object is initially moving to the right with a speed of 1m/s(NOT at rest), and the final speed is 3m/s, the KE change in frame O' = 4J, which is different from that in O. We know that in this simple case the KE change must equal to the work done on the object, and this must be true in both frames. Thus, we conclude the work done is different in different frames. This is not difficult to understand as in frame O', the object is moving with a greater average speed when the force is acting on it. Since the time that the force applied on the object must be the same in both frames, the displacement of the object in frame O' is indeed larger and the resulting work done is also larger.

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