Hi Rich,
Can you have another look at the phone model (I'm not sure what a 7940E is)? It sounds like whatever phone you are using must be a Class 3 device which requires 15.4 Watts. Here is an excerpt from a good doc on 3560-48 switches;
"Taking advantage of Cisco Catalyst Intelligent Power Management, the 48-port PoE configurations can deliver the necessary power to support 24 ports at 15.4W, 48 ports at 7.7W, or any combination in between."
PoE
Maximum power supplied per port: 15.4W
Total power dedicated to PoE: 370W
Here is a link to the actual doc;
http://www.cisco.com/en/US/products/hw/switches/ps5528/products_data_sheet09186a00801f3d7d.html
I do know that the 7941G-GE and the 7961G-GE are class 3 devices and would work as you have described with the 3560-48 switch.
Hope this helps!
Rob
Please remember to rate helpful posts....