MC, It's a matter of vector algebra, adding the various force vectors results in a net side force that can only pull the stylus toward the spindle (in the case of an overhung tonearm), because the stiffness of the arm wand prevents movement in the actual direction of the major net force, which is toward an ever-moving point that is always pointed to the rear but to the inside of the pivot (with a pivoted, overhung tonearm). With an underhung tonearm, the direction of the side force actually changes from pulling the tonearm inward to pushing it outward, after the stylus passes through its single null point, where there momentarily is zero skating force.

"Each and every point on a circle is the same distance from the center. Therefore the vector pointing towards the center is zero." It's not that the two statements are wrong. It is that the two statements have nothing to do with each other. Moreover, an LP groove is actually spiraling toward the spindle or the label, so each and every point is NOT the same distance from the center. And there is a net vector force toward the spindle; we call it the skating force. (I know we agree on that, but you seem to lose sight of it once in a while.)

The ball on a string goes off into space on a straight line tangent to its circular orbit, when you let go, because you were applying a force that kept it circling, until you let go of the string. That is called a centripetal force. Because as Newton tells us, "every object persists in its state of rest or uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it".

You wrote, "The motion of each point on the circle breaks down into a vector that is pointed straight ahead, ie tangentially, and straight towards the center. Each and every point on a circle is the same distance from the center. Therefore the vector pointing towards the center is zero." What?

The reason why overhung tonearms can never have zero skating force can be shown by the Pythagorean Theorem. As you say, tangency to the groove is what we are talking about, but we need tangency to the groove where the friction force generated at the stylus tip has a vector that passes back through the pivot point. Then and only then do we have zero skating force. Consider an underhung tonearm with no headshell offset angle that can achieve zero skating force at its single null point. In that one moment, the distance from the pivot through the tonearm/cartridge is one side of a right angle triangle (side a). The distance from the stylus tip to the spindle is another side of a right angle triangle (side b). And the pivot to spindle distance would be the hypotenuse of the right angle triangle, side c. Pythagorus told us that for any right angle triangle, c-squared = a-squared + b-squared. But if you have an overhung stylus, side a (tonearm effective length) is always larger than side c (P2S). So you can never achieve even a null point, let alone zero skating force, with an overhung tonearm, UNLESS you invoke a headshell offset angle. The founding fathers of cartridge alignment handed down to us a headshell offset angle, so as to achieve two null points across the surface of an LP. But they didn't give us any condition that satisfies what we need for zero skating force, because headshell offset per se causes a skating force.

"Each and every point on a circle is the same distance from the center. Therefore the vector pointing towards the center is zero." It's not that the two statements are wrong. It is that the two statements have nothing to do with each other. Moreover, an LP groove is actually spiraling toward the spindle or the label, so each and every point is NOT the same distance from the center. And there is a net vector force toward the spindle; we call it the skating force. (I know we agree on that, but you seem to lose sight of it once in a while.)

The ball on a string goes off into space on a straight line tangent to its circular orbit, when you let go, because you were applying a force that kept it circling, until you let go of the string. That is called a centripetal force. Because as Newton tells us, "every object persists in its state of rest or uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it".

You wrote, "The motion of each point on the circle breaks down into a vector that is pointed straight ahead, ie tangentially, and straight towards the center. Each and every point on a circle is the same distance from the center. Therefore the vector pointing towards the center is zero." What?

The reason why overhung tonearms can never have zero skating force can be shown by the Pythagorean Theorem. As you say, tangency to the groove is what we are talking about, but we need tangency to the groove where the friction force generated at the stylus tip has a vector that passes back through the pivot point. Then and only then do we have zero skating force. Consider an underhung tonearm with no headshell offset angle that can achieve zero skating force at its single null point. In that one moment, the distance from the pivot through the tonearm/cartridge is one side of a right angle triangle (side a). The distance from the stylus tip to the spindle is another side of a right angle triangle (side b). And the pivot to spindle distance would be the hypotenuse of the right angle triangle, side c. Pythagorus told us that for any right angle triangle, c-squared = a-squared + b-squared. But if you have an overhung stylus, side a (tonearm effective length) is always larger than side c (P2S). So you can never achieve even a null point, let alone zero skating force, with an overhung tonearm, UNLESS you invoke a headshell offset angle. The founding fathers of cartridge alignment handed down to us a headshell offset angle, so as to achieve two null points across the surface of an LP. But they didn't give us any condition that satisfies what we need for zero skating force, because headshell offset per se causes a skating force.