By simply shifting the center of gravity to one side and balancing the character properly, you can add the illusion of weight to the drawing.
In these examples I took some drawings (shown to the left) from an artist in the 1950's who wrote a book teaching how to draw cartoon characters (I believe that he worked for the Terrytoon studio). The drawings are rather awkward and reflect the studio style of that time period.
To the right, I've redrawn the character, making some modifications to the design as well, and pushed the poses to make them a bit more appealing. One of the main things you'll notice is the use of "S" and "C" curves throughout the drawings.
Another subtle thing is the use of tapered volumes on the arms, legs and torso. By making the upper part thinner and the lower part thicker, it tends to add weight to that part of the body. Do it all over and there will appear to be more of a pull of gravity on the character.
It all depends on the character design as well, in some cases, it may be that the arms are designed the opposite way, starting thick at the shoulders and tapering thin at the wrists.
Try to add this to your own drawings and see if you can notice the difference.