A YouTube channel, Vsauce, has claimed that the Internet weighs about as much as a rather large strawberry - 50 grams.
They base their calculation on the fact that electrons, however small, have mass while in motion, and that the functioning Internet involves the movement of many, many electrons.
However, that measurement of 50 grams is likely to be outdated, as the Internet has increased in size in the past few years and continues to increase exponentially with each passing year. The hosts of Vsauce also claim that a Kindle e-reader increases in weight, ever so slightly, with each book added. However, the gain is so miniscule that it's measured only by calculation—there being no scales sensitive enough for such a minute amount ("Internet 'Weighs the Same as a Strawberry,'" The Telegraph , Nov. 3, 2011).
