The "Laboratory" Frame Part IV

A part-inertial/part-laboratory experiment

Revisiting an earlier idea

When discussing the idea shown in the above figure, in my post of 25 April 2015, I assumed that the spring to be used to return the mass over portion B to C of the operating cycle would be a constant-force spring of force equal to twice the weight of the mass. Let's now look at this in more detail.

Does the mass-spring system act in full gravity?

First, recall that I have assumed that gravity always acts vertically, which it does, to an extremely close approximation, over the small time intervals of these experiments.

The value of the constant-force spring was calculated as:—

2W = 2mg = 2 × 10 × 9.80665 = 196.133 N.

Note that I used the same (standard) value for gravity, g = 9.80665 m/s² as I had used to calculate the behaviour of the mass between A and B. This is correct for a mass hanging freely from a suspension point and behaving fully "inertially", disconnected from Earth between B and C (as it was between A and B). However it would be possible to constrain the mass between B and C to radial movement only, e.g. in a vertical (radial) tube rigidly fastened to Earth. It could be argued that this should then modify the value to be used for gravity, to:—

g' = g - (Earth's centrifugal acceleration).

Hence, at the equator, g' = (9.80665 - 0.033916) m/s² , and the value of the constant-force spring becomes:—

2W = 2mg' = 2 × 10 × (9.80665 - 0.033916) = 195.45468 N.

So there would be a net gain of energy over the originally-calculated value, of:—

 (196.133 - 195.45468) × 0.271326 = 0.184046 joules, 

as the mass was returned from B to C, with the suspension point falling from B' to C' as before.

A net energy gain is calculated

This is a very low-power result, i.e. 0.184046J/10kg/8s  ≈  2.3 milliwatts of power per kilogram of active mass, over the operating cycle of 8 seconds. But nevertheless, it is a positive result as calculated.

I have set myself a simple "rule of thumb" — I will not even consider building any physical prototype machine that could not reach at least one watt of power per kilogram of active mass.

For now, I leave as an open question whether a gain like this could really be achieved in a system as discussed, where a mass is designed to behave truly "inertially" over part of its cycle, and is constrained into a laboratory frame for the rest of the cycle; or whether it just indicates some error, e.g. a breakdown of the assumption that gravity is always vertical. Two relevant points to bear in mind are:—

1. As soon as the mass is fired-off from Earth at point A, it no longer has any physical connection with Earth, and must act "inertially", albeit still under the influence of Earth's gravity.

2. The mass can certainly be constrained into a physical connection with Earth, forcing it to move radially in a laboratory frame over B to C, as discussed above. But gravity really acts radially, anyway! 

More experiments

There are obviously many more experiments that could be done, as developments of the ideas already discussed in this post and the three previous ones. I have done quite a few myself, including some that had a more clear-cut separation between "inertial" and "laboratory" behaviour than in the above introductory example. For example, a rotating-wheel gyroscope prefers to orient itself in an inertial frame as far as possible, rather than a laboratory frame. And the gyroscope can take other forms — such as the halter gyroscope, which many flying insects possess in some form. But to discuss these experiments further now would be to go too far out of order in what was always intended to be a roughly chronological-order blog.

Crane fly, with halteres visible behind the wings
Image from http://en.wikipedia.org/wiki/Halteres