Seanie

Good question. IF I were unaware of Newton's laws of motion then I may suggest that kinetic energy of m2 is caused by m1 taking on inertia (& therefore acceleration) in the x direction and that this inertia is given to m1 by virtue of the rod's rotation and since this happens without the need of extra torque being given to the rod then the inertia, acceleration and kinetic energy arise from the rotation itself. But since I am aware of said laws of motion then I can't seriously propose that.

Actually I was thinking of the rail being beneath the rod but I don't think it makes any difference whether it is above or below. I don't have a method in mind. All we need to know is that the rod is rotating at some arbitrary angular speed. I'm not sure what you mean. The only reaction forces I see at m2 are the forces against the sides of the rail which constrain it. m2's motion in the x direction is caused by the centrifugal force acting on it from m1's rotation, or if we consider centrifugal force not to be real then it is the inertia of m1 in the x direction which is transferred to m2 to make it move along the rail. The rod's rotation gives rise to inertia of m1, hence its motion in the x direction (& thus to m2 in the x direction also) and so the whole rod is in effect moving in the x direction as it rotates. As I understand it this motion in the x direction is not caused by a real force, so there is no reaction force to it. Perhaps you can show me where I am wrong here.

Sorry for not clarifying that, they are not all in the same plane. let the rod including m2 be in the same plane as each other and m2 is mounted in the rail just beneath it. So rod rotates just above the rail. Let me know if that is still not clear.

because the SHM applies to m2's linear oscillating motion along the x-axis and the constant angular speed applies to the rod's rotation about m2. It doesn't matter how.

Description of the set-up as given in the uploaded file: Consider a rod of length r, viewed from above, one end of which is called m2 and the other end is called m1. m2 is pivoted in a linear bearing so it can freely slide in a straight line in both directions as well as being free to rotate either clockwise or anticlockwise about m2 as the centre of rotation. The linear bearing lies along the x axis. At m1 there is a point mass of m1 kilograms. The rod is given an initial momentary torque to cause it to rotate clockwise about m2. Assumptions: Other than the force which produced the initial torque there is no subsequent force or torque given to the system from outside. The whole assembly operates in a horizontal plane so is unaffected by gravity. There is no friction. Everything has zero mass except for m1. The linear bearing is long enough to accommodate any range of motion which m2 may undergo. My analysis: Due to m1's rotation m2 should exhibit simple harmonic motion symmetrically about a point on the x-axis, let this be the point (0,0) in the Cartesian plane. According to Newton's first law of motion the rod should continue to rotate at it's initial angular speed of w1 (omega-1) since (as far as I can tell) there is no torque opposing its rotation. Let theta be the angle the rod makes with the y-axis so that when the rod is in line with the positive x-axis theta = pi/2 radians. m2's motion: It seems to me that the position of m2 is given by x=rcos(theta). Therefore m2's speed, v2, is the rate of change of x with time, which is -rw1sin(theta). Therefore the magnitude of m2's acceleration, a2, is the rate of change of v2, which is -r(w1)^2cos(theta) (i.e. minus r omega1 squared cos theta). Therefore the magnitude of the force, F2, acting on m2 is m1a2. Therefore the power, P2, in m2's motion is F2v2. The kinetic energy in m2's motion, E2, can be calculated from the kinetic energy formula of 1/2 mv^2 which gives the same result as getting the integral of P2. Discussion & request for help: If the analysis given above were correct that would mean energy is not conserved in this system since there is a quantifiable power output even if it is constantly changing in direction and magnitude, even though power is not being input to the system. Assuming therefore that the physics used here is flawed, then why and how exactly is it flawed since it is apparently the standard Newtonian mechanics that I know? Also exactly what is the correct physics for this mechanism? The answer to that must among other things be able to show exactly how torque arises which opposes the rod's rotation since it seems clear that as long as the rod rotates then it will provide power to m2's motion. Thanks in anticipation.

For some reason the file with the concept details will not upload to my post so you can view it at https://ibb.co/mHPKmRK instead.

Please see the attached "concept" pdf file. If the mechanism does not generate energy then there should be physics which can clearly show that without it being assumed in the method of analysis used. Included in the analysis should be clearly shown details of a torque opposing the rod's rotation since the absence of such a torque would mean that the mechanism continues to operate indefinitely and in doing so continues to power the motion of m2 indefinitely. If it does produce energy then energy is not conserved. More could be said about this interesting mechanism e.g. acceleration of m1 in the x direction without a force being applied to it in that direction, the force on m2 in the x direction without there being a reaction force, how rotation produces inertia, implications for Newton's laws of motion and so on. Thanks for your interest.