I will phrase this question as best I can. If you need clarification I will do my best. The question relates to a pendulum. For example I have a 12" string with a 2 oz metal ball on the end. I assume it has a certain degree of hysteresis but my question only relates to one issue. The ball wants to go in a straight line but the string keeps it on a specific arc. The string has no stretch. the speed of the pendulum determines the tension on the string. Where does the energy go used to keep the string tight? Is this force a significant part of the hysteresis?
If the string has no stretch, then no energy is needed to keep it tight. If instead the string had a small but nonzero amount of stretch, then it would take a small amount of energy, and that energy would be stored in the string.
The energy doesn’t “go” anywhere.
It becomes potential energy stored in the string (just think of the string as a spring).
No energy is expended once the string is stretched.
Eventually, all the energy in the pendulum becomes heat.
“hysteresis” means that the present state of the ball and string can’t be described by just it’s position and speed.
Most forms of energy loss are not hysteresis. Hysteresis does not have to include energy loss. (Hysteresis may include energy loss, but that’s only because everything includes energy loss.)
You can use a pendulum system to create hysteresis in a larger system. For example, every time I walk past a clock, I could wait until the second hand points to 12. But the pendulum itself is not exhibiting hysteresis.
If you have a brass pendulum, and it is gradually wearing out it’s bearings, that wear could be a form of hysteresis. You look at the pendulum, and today it’s in exactly the same position at the top of the swing, at exactly the same speed (zero), but today it’s stuck. It’s not a pendulum any more: as a result of it’s past history, it’s worn out.
But your question seem to exclude most forms of hysteresis. Which is no surprise: pendulum systems were chosen as clocks for their lack of hysteresis.
Title changed to add more detail.
The speed of the pendulum does affect the tension on the string, but the tension is also a function of gravity. (Get on a playground swing - at the top of your swing if you don’t reach horizontal, there is still tension on the swing’s chains. Or imagine the pendulum held up at the start position resting against your finger. There is still tension in the string, but some is also pushing against your finger Do the Force vector diagram.)
The total energy of the system is constant, ignoring details like friction loss and heat loss, air resistance, etc. You can do vector analysis to show that when the ball is at the top of its swing, it has added potential energy - i.e. raised up in a gravity field. When the string is vertical, there is no extra potential energy, it has been converted to kinetic energy, the ball is moving.
The additional tension on the string is determined by the velocity of the ball. To travel in a circle (or arc) requires a centripetal force directed to the center of the circular path (i.e. along the string). The amount of this force is determined by the velocity of the mass, so is obviously not constant.
Recall - F=ma p=mv but E=Fd and these have equivalent formulas for rotational force, momentum and energy.
Work/Energy is force times distance the mass is moved by that force. If the rotational force does not change the radius then no work is done. A weight on a string is no different than a spinning globe or rod when it comes to rotational Force and energy - just spinning is angular momentum, but does not use or produce energy as long as the angular velocity is constant. The total energy expended is due to gravity - pulling the mass down from a height. That energy translates height(potential) into kinetic (velocity) and then back into potential again up the other side.
(Imagine instead of a solid string, a spring. Then, the spring would elongate as the weight headed down to vertical, and contract again on the way up the other side. This stretching Force is a function of mass, velocity and Hooke’s constant.)
So where does the energy go? It translates back and forth between kinetic and potential energy.
Oh, and if you do have a springy string, that can hold a significant amount of elastic energy, the energy will be shared between two oscillatory modes: The pendulum mode, and the spring mode. If the two have similar frequencies, then energy will be transferred back and forth between those two modes, with the weight sometimes bobbing up and down, and sometimes swinging back and forth. Fun to watch, when it happens.
Glad you commented on the stretchy string, I was wondering about that also.