Friday, June 19, 2009

Generalizing Capacitors And Inductors To Include Spinning Wheels

Let us now consider a wheel spinning in its carriage and a capacitor; their workings and similarities.

A capacitor is a temporary storage device for electrical energyA flyweel is a temporary storage device for mechanical energy
A given capacitor has a maximum operating voltage beyond which the capacitor will discharge spontaneouslyA given flywheel has a maximum operating angular velocity beyond which it will break apart due to stresses with in the material
A discharging capacitor shows a continuous drop in its potential differenceA discharging flywheel shows a continuous drop in its angular velocity
The capacitance is determined by the surface area of plates and the permeability of the medium between the platesThe moment of inertia is determined by the mass distribution of the flywheel
Charge Q = C*VAngular Momentum L = I*ω
The energy stored in a capacitor is given by E = (1/2)*C*V2The energy stored in a flywheel is given by E = (1/2)*I*ω2
Charge plays the same roleas Angular Momentum
Capacitance plays the same roleas Moment of Inertia
Potential Difference plays thesame role as angular velocity

Accelerated Spinning Wheels and Inductors

Flywheels can also act in more unusual ways reminiscent of inductors under slightly different conditions. Such an arrangement is shown in the figure above.

When the flywheel is suspended in a carriage and the carriage is offset about a Y-axis as shown in the figure, the inductive condition is invoked as follows: The spinning wheel spins at a fixed angular velocity. The carriage is moved about the Y-axis with a torque that changes in time.

The existence of a significant rate of change of torque is a necessary condition to harness a spinning wheel to transfer energy via the inductive process.

Consider the behavior of an induction coil with a steady current through it. It resists the change of an existing current in the coil. Given that charge is analogous to angular momentum (from our discussion above),

Q(charge) ~ Lm (angular momentum)

We diferentiate this once wrt time to get

dQ/dt (current) ~ dLm/dt

Now, dQ/dt is nothing but current ( rate of change of charge with time).
And since dLm/dt is nothing but rate of change of angular momentum, we can write it as I*dω/dt=> I*α (ang acceleration).

That is, we get a NEW analogy
i ~ α
This adds to the previous three analogies regarding charge, capacitance and potential difference we made just now, i.e.

Current plays the same role as Force (= Moment of Inertia * Acceleration)

The characteristic formula for an inductor is its voltage to current relationship in time. An inductor undergoes self-inductance only in the presence of a varying current, i.e. a second derivative of charge over time i.e., di/dt.

now if i ~ α, in order to find ANOTHER NEW analogy, we differentiate it one more time to get

di/dt ~ dα/dt

The analogy above suggests therefore that the mechanical phenomena equivalent to inductance is given by the second derivative of angular momentum over time.

(Now, the first derivative of angular velocity is angular acceleration. The second derivative is therefore the derivative of angular acceleration).

Thus the amount of counter-angular-velocity and counter-angular-acceleration (since the development of the angular velocity will happen at a rate of acceleration that can be measured by measuring the rate of change of angular acceleration of the applied force) developed by a flywheel depends on the imposed rate of change of angular acceleration.

Just as the characteristic formula for an inductor is its voltage to rate-of-change-of-current relationship in time, the characteristic formula for such a spinning wheel would be its precessive angular velocity to its rate of change of angular acceleration relationship in time (taking into account the Moments of Inertia of the structures involved).

An inductor undergoes self-inductance only in the presence of a varying current, i.e. a second derivative of charge over time i.e., di/dt.

Thus, in analogy with the equation Vcounter(t)= L*di/dt for inductor coils, the equation

ωprecess (t) = (Inductance of Spinning Wheel)*dα/dt

represents the characteristic formula for a spinning wheel freely rotatable about a perpendicular, offset axis. Thus, just as voltage leads current in inductors, the precessive velocity leads the angular force (acceleration) in an inductively suspended spinning wheel.

Further, just as the counter e.m.f causes a counter current to flow through the inductor, so also in the flywheel the counter-precessive-angular-velocity would produce a counter-angular-force (acceleration). Understanding this analogy is the foundation for an alternative explaination of gyroscopic action of flywheels.

In a circuit including an inductor, we apply current and receive voltage.
In a gyro too, we apply torque (gravity pulling down on the wheel) and receive angular velocity (precessive motion), instead of receiving angular acceleration as we should, if we were to apply Newton's Second Law.

All this also implies that as a toy gyro spins and whirls, energy is being given away to the gravitational well at a measured rate as the gyro winds down. (unlike conventional explanations which insist that the energy is simply dissipated via bearing friction - now this transfer rate is very small, which is why it would appear that something minor like bearing friction is involved. However that is simply the easy explanation that got us into all this trouble in the first place.)

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