Gain, volume and output power

In addition to the thread I recently started about what the difference is between slew rate and rise time, I would like to have some explanation about the following subject: what is the relationship between gain, volume and amount of power that an amp delivers. Let's say an amp is capable of delivering 200wpc into 8 Ohms. Is this with the volume knob wide open or doesn't it relate to the position of the volume knob?
If you're talking about a basic power amp with a passive volume control on the input then wide-open relates to full volume. If you're talking about an integrated amplifier with and active input stage then the amp will probably be pushed to full volume/clipping well before the maximum volume setting is reached. The point at which you reach full volume will depend on the output voltage of your source and the recording level of the particular recordings being played...
It boils down to some simple math.

An amplifier takes an input voltage and multiplies it. The amount the amp multiplies by is the gain. If your amp outputs 50 volts with an input of 1 volt and then the gain is 50.

Vout/Vin = gain

50V/1V = 50

or this can be rearranged to give

Vout = Vin x gain

Your example amp has a maximum output of 200 watts into 8 ohms. If you take the 200, multiply it times 8 and then take the square root of that, it will tell you the maximum voltage your amp can produce. In your case that's 40 volts.

voltage = square root of (power x ohms)

A typical power amp will have a gain of around 20 so to get 40 volts out you would need to put 2 volts in.

Vout = Vin x gain

20 x 2V = 40V

It takes 2 volts to get max power so the sensitivity of the amp is said to be 2 volts.

If it had a volume control this would be with the volume all the way up.