Hence the vector of the centripetal force and the vector of the centrifugal term are at every distance to the center equal to each other in magnitude and opposite in direction, so those two terms drop away against each other. It is only in very special circumstances that the vector of the centripetal force and the centrifugal term drop away against each other at every distance to the center of rotation. This is the case if and only if the centripetal force is a harmonic force. In this case, only the coriolis term remains in the equation of motion. Since the vector of the coriolis term always points perpendicular to the velocity with respect to the rotating coordinate system, it follows that in the case of a restoring force that is a harmonic force, the eccentricity in the trajectory will show up as a small circular motion with respect to the rotating coordinate system. The factor 2 of the coriolis term corresponds to a period of revolution that is half the period of the overall motion. As expected, the analysis using vector notation results in a straight confirmation of the previous analysis:  The spring is continuously doing work. More precisely, the spring is oscillating between doing positive work (increasing the weight's kinetic energy) and doing negative work (decreasing the weight's kinetic energy). In the section 'Energy conversions in rotational-vibrational coupling' the dynamics is followed by keeping track of the energy conversions. It is often pointed out in textbooks that the increase of angular velocity on contraction is in accordance with the principle of conservation of angular momentum. Since there is no torque acting on the circling weights, angular momentum is conserved. However, this disregards the causal mechanism, which is the force of the extended spring, and the work done during its contraction and extension. Similarly, when a cannon is fired, the projectile will shoot out of the barrel towards the target, and the barrel will recoil, in accordance with the principle of conservation of momentum. This does not mean that the projectile leaves the barrel at high velocity "because" the barrel recoils. While recoil of the barrel must occur, as described by Newton's third law, it is not a causal agent. The causal mechanism is in the energy conversions: the explosion of the gunpowder converts potential chemical energy to the potential energy of a highly compressed gas. As the gas expands, its high pressure exerts a force on both the projectile and the interior of the barrel. It is through the action of that force that potential energy is converted to kinetic energy of both projectile and barrel.