A failing power supply or power switch are reasonable guesses. But if the power supply had failed, then the computer wouldn't power up at all; if the switch had failed, then the switch would never power up the computer.
I suspect there's a less catastrophic reason for these symptoms. Let's define the problem more precisely; Is the problem that you hit the power switch, the chime sounds, but the display does not light up? Or is the problem that you hit the power switch, and nothing happens at all?
If the first one is the case, then the problem could be a weakened or dead PRAM battery. PRAM (parameter random access memory) is a special body of data stored in RAM that keeps important settings when the computer is powered down, including screen resolution and time. Traditionally, these settings were maintained in PRAM when the computer was powered down by means of a small battery socketed to the logic board. Beginning in 1999, the PRAM battery was supplemented by trickle power from the AC connection, or the main battery in portable devices. Recent laptops have eliminated the PRAM battery entirely.
The Early 2009 iMac still includes a PRAM battery - a BR2032 lithium ion coin-type disk, like the ones used in small remote controls, timers and Bluetooth headsets. After five years, your original PRAM battery is probably dead. The first visible signs of a dead or dying PRAM battery are:
- Incorrect time settings on cold startup (the internal clock resets to dates long past, such as January 1 1970)
- Black screen after the startup chime
A lot of people set their system time to synchronize with an online time server. As soon as the computer establishes an Internet connection, the time is corrected. This can conceal a dead battery for a long time, unless the dark screen symptom appears.
When the computer starts, it sends a sync signal to the attached display. If the signal is not received, the display remains dark, although the operating system is loading. Sometimes when the user force-restarts the computer after startup, enough static charge will be retained in the system to supply the sync signal on the following startup.
If this is the situation, it's much more convenient for use to eliminate the problem by replacing the PRAM battery. Instructions for the process are here:
iMac Intel 20" EMC 2266 PRAM Battery Replacement
As this repair involves removing the display and opening up the computer, this is also a good opportunity to upgrade the hard drive.
iMac Intel 20" EMC 2266 Hard Drive Replacement