1. The problem statement, all variables and given/known data Not a homework or coursework question, but given the simplicity of the problem I feel that this is an appropriate subforum. Consider a person spinning a rock on a string above their head at a constant angular velocity, walking away from the observer at a constant linear velocity (ignore gravity for simplicity - consider it a 2D problem as viewed from above). The goal is to determine the total kinetic energy of the rock in the observer's frame. 2. Relevant equations Typing this on my phone so apologies for the lack of LaTeX I = mR² (ignoring the string) Angular kinetic energy = (1/2)Iω² Linear kinetic energy = (1/2)mv² 3. The attempt at a solution In my mind there are two ways to solve this, but they don't agree with each other: The first way is to simply consider the problem as a superposition of the rotational and linear components, and sum their energies: E_k = (1/2)mv² + (1/2)mR²ω² Which gives a time-invariant answer The second way is to consider the motion parametrically. The motion could look something along the lines of (letting the walker's velocity wrt the observer be u and the velocity of the rock in the observer's frame be v): x(t) = Rcos(ωt) + ut y(t) = Rsin(ωt) Taking time derivatives of each coordinate yields x'(t) = -Rωsin(ωt) + u y'(t) = Rωcos(ωt) The kinetic energy would then be E_k = (1/2)m(x'(t)² + y'(t)²) Which for u ≠ 0 is not time invariant. After considering how linear and angular velocity components can "cancel" each other if the linear velocity is not normal to the plane of rotation, a different answer is reached. I can't convince myself which should be correct (if either). The problem I have with the first solution is that it doesn't consider how the two motions can cancel, and the problem I have with the second one is that the energy shouldn't change with time. For the energy to change, work would have to be done (and the energy would have to go somewhere) but the only force acting on the rock is the tension of the string, which doesn't move relative to the rock and shouldn't be doing any work. I think that the second solution is not considering some sort of virtual force which would explain how energy is periodically removed and then restored, but I'm not sure why a virtual force would appear in what I think is an intertial frame (but it's possible that my understanding of inertial frames is wrong as well). The rock is of course accelerating with respect to the observer, but if you choose u = 0 then the two solutions yield the same answer and no virtual force is required to describe this scenario (despite the rock still accelerating in the observer's frame). I'm in 3rd year engineering (and ashamed that I can't answer this) so I have a reasonably thorough math background - if an adequate answer requires some university-level math then don't hold back. Thanks in advance for any responses!