[Jeff]

Bernt

In your formula, the work output is zero because of the way the formula is structured. That's why I sometimes distrust the input of physicists (like nmgolfer) who are capable of deep mathematical expositions based on mathematical formulas. The "real" issue is not the accuracy of the formula, the "real" issue is it's relevance. The most important question is what's the best perspective to look at a problem, and then one has to decide which formula to use in that situation.

In the situation of centripetal force, if the force doesn't provide the tangential force needed to move a mass at a certain speed a certain distance, then one shouldn't be using a formula that 'a priori' uses those requirements (speed and distance) to calculate work output. Secondly, the idea of a centripetal force always being at 90 degrees to the mass is only a mental concept, and it obviously results in zero work output according to that formula.

Consider the example I gave of person B applying a force to deflect the mass (that was being pushed in a straight line direction by person A). Presume that he doesn't apply a force at right angles to the moving mass, but presume that he stands at an angle to the mass - as described in this next example - and pushes in the direction of the arrow.



If you look at the angle that he is pushing, one can imagine that the object will not travel in a straight line path and that it will be deflected slightly leftwards. The amount that it will be deflected leftwards depends on the magnitude of person B's push-force relative to the magnitude of person A's push-force. If person B's push-force far exceeds person A's push-force, then the degree of leftwards deflection will increase. Note that person B's push-force will make the object travel faster, because a component of the push-force is working in the same direction as person's A's push-force. In other words, from a conceptual perspective, one can describe person B's push-force as having two directional components - a vector component that works in the same direction as person A's push-force and helps increase the speed of movement of the mass in person A's straight line direction, and a vector component that works at 90 degrees to person's A's straight line path, and that causes the mass to be deflected slightly leftwards. Both vector components are doing work.

Now consider another example.



Note the direction of person B's push force. It is directed somewhat backwards. If one dissects person B's push-force into two directional components - one vector component will work in direct opposition (180 degree angle) to person A's push-force and that will slow the speed of movement of the mass. The other vector component will conceptually work at right angles to person's A direction of push-force and that will cause the object to be deflected leftwards. Both vector components are producing a work output. The amount that the object is deflected leftward depends on the magnitude of person B's push force relative to the magnitude of person A's push force.

Hopefully, you will understand what points I am trying to make.

1) The first point is that the force (exerted by person B) that deflects the object leftwards is a "real" force that requires energy, and one has to rationally conclude that the force is doing work by deflecting the object.

2) After the object has been deflected, one can look back at the circular path that was transcribed on the ice rink and one can 'a posteriori' theorize as to what "force" resulted in the path being circular rather than straight line (towards destination D). One can simply conceive/theorize that a deflection force was present that caused the path to become circular-shaped. One can conceive that the "force" has centripetally accelerated (deflected) the object - defined simply as a "force" that causes an object (that already has enough energy to move in a straight line direction) to follow a circular path rather than a straight line path. In one's mental conception, one can conceive that the "force" is directed towards the center of a hypothetical circle, which means that the "force" is operating at 90 degrees to the circular path transcribed on the ice rink. However, this "force" and its 90 degree directional angle relative to the final arced path transcribed on the ice rink is merely a mental construct. In reality, there was only one force exerted by person B and it was in the direction of the red arrow, and the red arrow is not perpendicular to the circular path's arc.

Jeff.