Series-parallel Springs

Exchanging potential and spring-stored energy

Suppose we allow a weight that is suspended from a tension spring to fall slowly ("isothermally") against a load. The spring is initially unstretched, and the falling weight delivers energy into the system until the spring force equals the magnitude of the weight. Then everything is in equilibrium, with no further movement.

Is it then possible, without adding any more energy, to extract all of the energy stored in the spring, and to raise the weight? The answer is yes. Of course the weight won't be raised back to its original height, because of the energy it already delivered, but we can still convert all of the spring stored energy back into potential energy of the weight. We can do this by initially constructing the spring as a pair of equal springs connected in series, and then reconnecting them in parallel.

Two equal springs connected in series, from a weight to the center of a wheel

The above silux model shows a series-connected pair of springs attached to a 4kg weight (red) which has fallen slowly as the wheel turned through 90 degrees, from an initial horizontal position with unstretched springs at 0.2m radius, to the vertical position shown, at 0.3180m radius. Each spring of k = 667N/m was stretched from 0.1m to 0.1590m.

If the low-mass spring joiner (purple) is now attached to the wheel, and the still stretched top spring is reconnected between the joiner and the weight (i.e. in parallel with the bottom spring), the springs will contract back to their unstretched length, pulling the weight up to a smaller radial distance of 0.2590m, where it can be caught and attached to the wheel.

The operating cycle is then finished by returning the weight back through 0.2590m to the horizontal position. Obviously, this will require energy of mgh = 4 × 9.80665 × 0.2590 = 10.16 joules. So, how much energy was delivered as the weight fell?

Graph of wheel torque vs wheel angle

Torque vs angle analysis

From the silux model, with the wheel turning at a very slow 0.01 radians/sec (to avoid any problems with centrifugal forces etc), we get the torque vs angle curve in the graph above. Integrating this curve, we get 10.15 joules of energy delivered, i.e. within 0.1% of the 10.16 joules required to return the weight.

A critic might say "O.K, you've just spent a lot of time and effort proving that there's no excess energy. That was obvious anyway from the conservation of energy principle." But to me, that was the point of this exercise, to carry out a detailed analysis for a particular case, and thereby get a fairly deep understanding of it.

From gravitational to inertial mass

More could be said about series-parallel springs, but I've decided not to say it just yet.

I have now finished my posts on devices that only exploit gravitational mass. Next time:— on to the second possibility mentioned in my post of 14 April 2014, i.e. "inertial" mass.