Rotation is a separate magnitude to translation. When a pressure is applied to a body it is within the context of an inertial equilibrium system holding the object in its static or dynamic situation. Thus an arbitrary pressure arising in such a system will generate arbitrary trochoidal motion .
When- sphere or a circular disc is analysed, often the inertial system is wrongly characterised. Firstly a rotation is observed and then assumed to have arisen by a tangential impulse acting in opposition either to another tangential impulse or a inertial central force.. These opposing force descriptions are defined as torque. They are also called a moment or a couple. By such a disposition all rotational motion is modelled, and indeed believed to be generated.
The generation of rotational motion is so varied that to assume it is a special case of moments and couples is misleading. This is especially clear when one leaves the cosy world of rigid body motion and enters a fully fluid domain.
In the inertial frame, when equilibrium is disturbed, then inertial pressures appear as if by magic to restore or retain equilibrium. However, the effect is propagative and sequential and if equilibrium is restored it is often through oscillation or damped oscillation.. The stability of an equilibrium is indicated by these outcomes. However unstable systems reach a certain level and then run away. In such cases we might regard the inertial pressures as resistive.
For a rigid body rotation the resistive pressures may occur tangentially to a pressure which acts on the whole body. Because of this a torque or a moment acts relative to every other point in the body. However when the instantaneous torque has acted there is an instantaneous angular or momental acceleration. This angular acceleration increases until the angular velocit is such that the rigid body outpaces the initiating pressure. At that stage the motion of the rigid body is purely due to rotation an the pressure gradient is no longer able to keep up .mthis is no longer torqued motion because there is no resistive force what is happening is dynamic angular momentum. If the angular velocity increases beyond the ability of the tangential drag force then slippage occurs.
If the slippage does not occur then the angular acceleration now uses the frictional forces to drive the rotating body forward. There are other resistive pressures that now occur attempting to maintain a larger equilibrium.
The discussion above regards torque as impulsive only. A rotational motive is engendered by some torque, but this is not the only way to start a rotational motion or induce a rotational motive. Rotation exists independently of tangential torques!.
Rotational motion like all motions can be resolved along different axes, but that does not make these resolutions motives of the rotational motion..
What is torque? It is officially defined as the stopping force. How much force is needed to stop a turning wheel?
It is clear that the faster the wheel is turning the more pressure. But it is also true that the further from the centre the pressure is appliedthe greater the instantaneous tangential force that is required!
The turning wheel is mounted on an axle and is virtually free of friction. To stop it we have to apply a normal pressure to generate a tangential resistive one. The angular deceleration times by the mass of the wheel must be proportional to the stopping pressure. So why the use of I?
The technical problem is translating between 2 independent magnitudes, one rotational and the other translational.
If I take a frictionless pulley and attach a light unstretchable string to it for one revolution to the free end I attach a mass. Then I assume a gravitational force fiel that is constant acceleration, then I can imagine an accelerating tension acting at the point of attachment. This tension is in fact wrapped around the pulley so a normal force of 2 times the tension cts on the pulley where the string is in contact. As the acceleration is applied the variation will create oscillations in the Bering of the pulley. Wrapping the string round more than once will minimise this effect only if the strings mass acts as a damping mechanism.
Leaving these to one side I can assume a constant tension acting at the point of attachment with no frictional resistive forces contributing to the motion.the only resistive forces are compressive supporting the tension of the string around to the point of contact.
There are many assumptions made bout this tangential tensile force. It can not be eassumed to act on all the surface of the pulley or resistive forces will need to be considered all the way round and a constant accelerative tangential tension will mean a different action is being considered. It is like having one person pushing a roundabout compared to a million pushing it all at the same time.
There are other considerations, too, for example how the accelerating tension, transmits the motive around the curve, whether this is instantaneous or propagative.
So finally we make this approximating model and it shows that the radial distance of the point of attachment has an astonishing effect: the nearer to the centre this tangential acceleration is applied the greater the angular acceleration of the whole!. Thus the principle of the lever is seen to apply in the case of tangential and so instantaneous pressure if constant.
I can now propose that tangential acceleration is related to rotational or angular acceleration by the radius of application,
Firstly I wii define Twistorque as mass times angular acceleration.
€ = mæ
Then I can define tangential force as
Ft = €r
Finally traditional torque is defined as
Traditional torque therefore has only an instantaneous application to a pulley, but it can be mechanically engineered to produce constant acceleration, continuously or discretely as in the gear chain systems in a grandfather clock. Again a geared action is entirely different to a tension action., but a pendulum gear control can metronome it out.
Traditional torque is best used in oscillating systems, where the forces are aimed at being balanced. To use traditional torque to define stopping force for a freely rotating system is misleading , so to hide this they rewrite Ftr as Iæ
There is no concept of Twistorque until my analysis last year(2012), and even thn I was feeling my way.
Now I can use a Twistorque vector to represent angular force, especially using radians
The unit angular acceleration is
and the arc acceleration is
This is clearly the identity with the instantaneous tangential acceleration in a tension system, with the above caveats . And we can relate this to traditional torque acceleration by
where r is the radius of application of traditional torque acceleration.
To use the equations or identities, the idea is to match R and r
The angular acceleration is always defined in radians by an rc lengt on the surface of a sphere of unit radius. However, the arc does not have to be a great circle arc, and in fact could be a spiral.
The notion of Angle in space is to be generalised to the area of a shape cut out on a spheres surface, forming a " cone" at the centre of the sphere.