Just use the plain ol' class selector.
$('#my_id .my_class')
It doesn't matter if the element also has other classes. It has the .my_class class, and it's somewhere inside #my_id, so it will match that selector.
Regarding performance
According to the jQuery selector performance documentation, it's faster to use the two selectors separately, like this:
$('#my_id').find('.my_class')
Here's the relevant part of the documentation:
ID-Based Selectors
// Fast:
$( "#container div.robotarm" );
// Super-fast:
$( "#container" ).find( "div.robotarm" );
The .find() approach is faster because the first selection is handled without going through the Sizzle selector engine – ID-only selections are handled using document.getElementById(), which is extremely fast because it is native to the browser.
Selecting by ID or by class alone (among other things) invokes browser-supplied functions like document.getElementById() which are quite rapid, whereas using a descendent selector invokes the Sizzle engine as mentioned which, although fast, is slower than the suggested alternative.