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How heavy is 0.00000780 tons?

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It's about as heavy as an Eye (Human).
(adult)
A fully developed eye (in a person over the age of thirteen), weighs about 0.00000780 tons. The pupil of the eye varies depending on the amount of light it is exposed to, but typically measures about 4 mm across.
It's about as heavy as a sheet of Paper.
(for US Letter, a.k.a. ANSI A; 215.9mm by 279.4mm (8.5 in x 11 in); 20lb)
The weight of a ream (500 sheets) of 0.01 tons, 8.5 inch x 11 inch paper is 0.002 tons, each sheet weighing 0.00000600 tons. Paper of this size has a thickness of 0.004 caliper (0.1 mm).
It's about one-and-one-fourth times as heavy as a Quarter.
In other words, 0.00000780 tons is 1.2480 times the weight of a Quarter, and the weight of a Quarter is 0.80130 times that amount.
(United States quarter-dollar coin) (a.k.a. Quarter dollar, a.k.a. America the Beautiful quarter, a.k.a. two bits)
The America the Beautiful series quarters, which began to enter circulation in 2017, weigh 0.000006250 tons each. In answer to the common riddle, there are 119 ridges around the edge of the coin in its current design.
It's about three-fifths as heavy as a Battery (AAA).
In other words, 0.00000780 tons is 0.590 times the weight of a Battery (AAA), and the weight of a Battery (AAA) is 1.70 times that amount.
(Alkaline)
An "AAA" size battery weighs 0.0000130 tons. Batteries are named by the American National Standards Institute (ANSI) Committee C18 and AAA batteries were given their name because they were smaller than the "AA" and "A" size batteries which were invented first.
It's about one-and-seven-tenths times as heavy as a Die (Dice).
In other words, the weight of a Die (Dice) is 0.590 times 0.00000780 tons.
(for six-sided, 16 mm, rounded cube)
A die, of the size most commonly used in board games and casinos, typically weight 0.00000450 tons. The use of dice and dice-like tools predates written history, with the oldest known examples belonging to a 5,000-year old backgammon set uncovered during an archaeological dig in modern-day Iran.
It's about half as heavy as a Compact disc.
In other words, 0.00000780 tons is 0.470 times the weight of a Compact disc, and the weight of a Compact disc is 2.10 times that amount.
(a.k.a. CD)
A compact disc typically weighs 0.0000170 tons. Data is stored on CDs by encoding into the track a series of tiny pits — each between 3.5 and 850 µm (0.00014 and 0.033 in) in length.
It's about three times as heavy as a Penny.
In other words, 0.00000780 tons is 2.80 times the weight of a Penny, and the weight of a Penny is 0.360 times that amount.
(United States) (formally one-cent coin) (Union shield design, 2010-present)
The weight of a United States penny is 0.00000280 tons. According to a 2010 report by the United States Mint, the cost of manufacturing and distributing a penny is $0.0179 — more than its face-value.