Why are low impedance speakers harder to drive than high impedance speakers

I've heard it described like this:  remember that the amp is not just pushing against a spring in one direction, but rather pushing and pulling back and forth.  Less impedance, e.g. a lighter spring in your analogy, means more slop, more wiggle room, that the amp must now cope with in order to control the woofer.

Almarg, please weigh in and set us all straight.  ;-)

It's about the amp.  Amps like to put out at 8 ohms.  The mechanical analogy isn't a good one.  Instead, think about your past gf's.

A low impedance speaker requires greater current from the amplifier to result in the same voltage at the output.

I ain’t Al but one of the few things I sort of understand about electronics is Ohm’s law.

From Wikipedia
Ohm’s law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equation that describes this relationship:

I (current; amps) = Voltage divided by Resistance (ohms)

At the risk of embarrassing myself, as resistance decreases the current required for a given voltage goes up.

As an example:
1 volt/8 ohms = 0.125 amps;
1 volt/6 ohms = 0.167 amps;
1 volt/4 ohms = 0.25 amps
1 volt/2 ohms = 0.5 amps

So, for a constant voltage, current requirements double going from 8 ohms to 4 ohms and quadruple going to 2 ohms. Current production is a more useful indicator of amp "grunt" than rated wattage (or so I think).

The other thing to remember is that for a given speaker design, resistance might not be constant across the frequency range. Low resistance dips occurring at different points can result in differences in how things sound between two amps with similar wattage ratings but differing in current output capability.

That’s what I think I know.