Why can't you just ground the scope or test instrument and measure those 120V volts? why is there need to "float" the instrument?
Because the chassis of the TV (or whatever) is live - connecting the scope to it is likely to result in a large BANG!.
It's an often discussed situation on here - and as a professional engineer I wouldn't consider using a scope that's earthed, it's far too dangerous.
People (including many on here) seem to think that earthing something makes it 'safe', this is totally untrue - it makes it 'safer' under some circumstances, but far more dangerous under many other circumstances.
What makes it 'safe' (for people who fully understand it), is understanding EXACTLY why earthing is and isn't used, and what you're dealing with.
To be fair it's pretty difficult to get a fatal shock from 240V, yet a number of people do it every year, and even from 120V as well.
I think the problem is I don't fully undertand what you guys mean by "isolated".
Why is the valve amplififier using transformer, isolated, if grounds are shared?
Why "isolated", if there's a shock hazard between any voltage point and ground?
What are we really after, when seeking "isolation"?
I'd really appreciate if anyone could help me understand it
Isolation means 'isolated from the mains' - not isolated from earth. An isolation transformer shouldn't break the earth connection.
The reasons for this are best explained historically, from back in the valve days. Old TV's and radios had just two core mains leads, and no transformer - the live and neutral weren't even marked!. The two wires entered the TV/Radio, one went to the rectifier (hopefully the live - 50/50 chance), the other was connected directly to the metal chassis (hopefully the neutral - again, 50/50 chance). So you had a 50/50 chance that your chassis (and every metal part of the unit) was connected directly to the live wire from the mains socket. So, earthed scope clip in one hand, TV/Radio chassis in the other - full 240V mains directly connected across your chest
In 'later' years, the electric companies complained about this practice (using half wave rectifiers), as if most were connected the right way round it mean people were only using the +ve half cycle of the mains, and this caused a DC offset on the mains. As a result manufacturers moved to using bridge rectifiers, this meant the chassis was at half mains potential regardless of the polarity of the wires - with similar consequences for your heart, or indeed your scope!.
So using an isolation transformer breaks the live and neutral from the unit, providing an isolated 240V (or 120V) supply, and in the case of an earthed chassis that should remain earthed.
A valve amplifier, using a mains transformer, is already isolated, so doesn't particularly need an isolation transformer.
As a professional engineer I'm perfectly happy to work with or without an isolation transformer, whichever the case you should consider
EVERYTHING is live and act accordingly.
Interestingly I was talking to a plumber we use at work (for installing gas or electric cookers), and he was telling me he went to a house to fit a new gas boiler, and there was no earth connection
ANYWHERE in the house
He obviously advised her that she needed the house rewiring (urgently), but she wasn't prepared to do so - however, he couldn't install the gas boiler without an earth, so he had to install an earth spike, earth the electrical work for the boiler, and earth bond the gas and water pipes. While the UK probably has the best electrical system in the world, there are still a few VERY old installations that have never been updated from many decades ago when specifications were completely different.