Jim has explained his version above, hopefully without starting a flaming debate which I have neither the time or desire to get into, I will attempt to define my version (OPINION);
I say that Cp and Cf are not the same force. In fact Cp used in the sense of rotary motion is not a force, it is static inertia, such as pushing on an immovable object Cp represents a fixed impediment that Cf cannot break through, if it does then you no longer have Cf or Cp in a radial path.
Cf is a force created by inertia, it can be measured and will vary with velocity only if it is physically contained by a barrier or containment that presents no force in its self. It is no different than trying to push your fist through the top of your desk.
How doe one measure the force from the desk top to your fist and what is creating this force if there is one? The weight on a string is applying tension on the string, there is no difference which end the tension is measured from as it is Cf pulling outward, there is no force pushing inward.
A weight riding on a wheel produces an outward force that is not balanced by an inward force, it is contained by physical means having no force other than inertial mass to keep it from allowing the weight to follow a straight trajectory.
Whenever Cp is present then you will also have Cf present, and visa versa.
True! but at static there is no Cf yet the physical means of Cp still remains.
Gravity pushes the Moon toward the Earth and gravity pushes the Earth toward the Moon. Do we have a different words for gravity depending on whether we are on the Moon or on the Earth? No! Then why do we need different words for Cf and Cp?
Is this a clerical error or are you serious? Gravity attracts.
I was taught that gravity pulled on the moon and the moon pulled on the earth (high and low tides) I have never heard of them 'pushing each other. So what keeps them from colliding? Cf with a little Am thrown in.
The fact that both are rotating while one revolves around the other may have something to do with it. But I am not going there.
Someone once said that we need to now which is which during calculations. This make no sense because both are tensions and tensions are always positive. The restraint always pulls inward while the weight always pulls outward. When the motion stops then the force also stops. The force is an artifact of the motion.
I agree that it makes no sense but my reasoning is; there is only one tension and that is the weight pulling outward. There is no tension pushing or pulling inward unless the mechanism is designed to do so, it is as static as the top of your desk.
Some people state that Cp (the inward pull of the restraint) is a real force while Cf (the outward tnedency of the weight) is a ficticious force. This conjures up thoughts of truth vs lies or real vs fake.
Cp is not an inward pulling force, it is a physical restraint. To gain a physical pulling force you must pull in, such as letting the string wind up around your finger.
As a final comment I see no reason to differentiate between Cp and Cf, so I simple use CF to mean that force that develops between a restraint and a weight moving in a curve path that is caused by inertial momentum of the moving weight.
I agree, there is no reason to differentiate as Cf is the only variable and the tension gradient is measured from a fixed point referred to as Cp.
Now, why the debate if I agree? Because I am chasing Cp as a variable, when you wrap the string up on your finger you are creating tension (force) and an elliptical orbit adding angular momentum and angular velocity which both in themselves is considered force.
For an example lets take a coopered wooden axle 8" in diameter, and tie a string and weight to it. Start the whole thing turning until Cf forces the string to become tight, stop the axle and let the string continue to wind. You are decreasing the radial path by an elliptical path diminishing at 25.1327" per revolution, That is Cp at work!
I leave it to the mathematicians to derive what happens if the string is released allowing not Cf but inertia to throw the weight in a straight trajectory at a right angle to the axle until the restraint once again becomes tight. If you have much weight you better have one tough axle.
Ralph