Centrifugal force in the loop?
Hello,
To better understand the forces at work in the loop, I watched YouTube videos. At least half of the videos attempt to provide a clear explanation of why F(G) = F(Z) at the top of the loop by vaguely explaining in a subordinate clause that it would be perfectly logical for the centrifugal force to completely offset the ball's gravity at that moment.
I just don't understand how anyone could seriously explain this in a physical-visual way using centrifugal force. I assumed that centrifugal force was an apparent force and only relevant in the system of the sphere, which is why it's so strange to me that easily half of it (and this isn't limited to the looping case) comes from centrifugal force.
I realize that the mathematics remains the same, whether it is centripetal or centrifugal force, but I simply cannot figure out how to clearly convey that the centrifugal force is a real force that pushes the ball outwards.
Which still doesn't answer the question for me as to why F(centripetal force) = F(G) holds at the highest point.
Am I wrong in what I said above?
Thanks in advance!
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Hello,
You're right. The centrifugal force is an apparent force. It doesn't act independently of the reference frame and doesn't fulfill Newton's third law. It's merely a useful mathematical model. Furthermore, at the highest point of the loop, the centrifugal force is NOT always equal to the force of gravity. This is easily explained by the fact that the force of gravity is constant, but the centrifugal force depends on the angular velocity. You stay in the loop and don't fall down as long as it mathematically holds that the centrifugal force is equal to or greater than the force of gravity.
The situation can be viewed either in a co-rotating (i.e. accelerated) reference system or in an inertial system.
1. In a co-rotating system (rotating of course not around the sphere's center, but around the loop's center), one must introduce a centrifugal force, as you mentioned, so that Newton's law F=m*a applies.
In such a reference system, the sphere is at rest, so the forces must add up to zero. In general, at the highest point
with
FN is the normal force from the looping track on the ball, and FG and FZ point in opposite directions. In the limiting case, where the speed is just high enough that the ball does not fall at the highest point, FN=0 and thus
2. In an inertial frame, there is no centrifugal force. However, the ball is traveling on an accelerated trajectory. At the highest point, only a normal acceleration occurs, which is a=v^2/r (centripetal acceleration, directed downward).
The equation
where in the limit FN=0 again applies and consequently
The "centripetal force" is not an independent force in the sense that it is due to an interaction such as gravity. In this case, the weight and possibly also the normal force act as a centripetal force. A centripetal force points toward the center of curvature, in this case the center of the looping track.