“Consider how easy it is to squeeze an empty soda can compared to how easy it is to crush a thin aluminum tube. If you cut the soda can open and curl the aluminum sheet into a smaller diameter it gets much harder to dent or deform. This has to do with leverage. If you take a wire or bar of metal it is much easier to bend, requires less force, if your hands are widely separated because you have more leverage. If the thickness of metal wall stays the same then larger cans are easier to crush. So assuming the thickness of the watermelon rind is about the same for small versus large melons it makes sense that the rubber bands have less leverage.
Also, the more you stretch a rubber band the more force it exerts. If the circumference of the big melon is twice that of the small one the same rubber band squeezes about twice as hard when stretched over the larger one. A combination of leverage and higher force makes the big melon take fewer bands to split.
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Try squeezing an egg from the ends the squeeze it across the middle. It is much harder to break across the ends.
The water melon is a fairly complex structure with a tough outer skin over a thick flexible rind covering a pulpy middle that has very little strength. Much of the drama in the video comes from the dramatic catastrophic failure of the outer layers leaving the weak interior material to be crushed and squirted about by the rapidly converging rubber