latex - includegraphic with variables in the name

This has the sweave tag, but what you show is not a complete LaTeX document with well delimited R code, but a chunk of hodgepodge of LaTeX plus R commands, including (maybe) the end of R code (@ line) but without any start of a R code chunk (<>>>= line), so it is difficult guess what are you doing really, but it seems unnecessarily complicated:

documentclass{article}
usepackage{graphicx}
begin{document}

<<objectnum,echo=F>>=
# Advanced method to set the variable:
graphicsrc <- 'example-image'
@

% Advanced method to use the variable as   filename of an image
<<echo=F,out.width="2cm",fig.cap="Example figure taking the variable name in R",fig.align='center'>>=
include_graphics(graphicsrc)
@

% Less advanced method 
begin{figure}[h]centering
includegraphics[width=2cm]{Sexpr{graphicsrc}}
caption{The same variable used in LaTeX code}
end{figure}

end{document}

BTW, as you can see in the example, you do not need to declare the extension of the image, except for disambiguation purposes (e.g., when you have foo.png and foo.pdf but you must use "foo.pdf" and not "foo.png").

On the other hand, do not see the use of [h] as a good practice. It was convenient only for this example.