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How heavy is 5.6700 grams?

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It's about as heavy as a Quarter.
(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 5.670 g each. In answer to the common riddle, there are 119 ridges around the edge of the coin in its current design.
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 9,000 grams, 8.5 inch x 11 inch paper is 2,000 grams, each sheet weighing 5 g. Paper of this size has a thickness of 0.004 caliper (0.1 mm).
It's about four-fifths as heavy as an Eye (Human).
In other words, the weight of an Eye (Human) is 1.30 times 5.6700 grams.
(adult)
A fully developed eye (in a person over the age of thirteen), weighs about 7.10 grams. 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 one-and-two-fifths times as heavy as a Die (Dice).
In other words, the weight of a Die (Dice) is 0.710 times 5.6700 grams.
(for six-sided, 16 mm, rounded cube)
A die, of the size most commonly used in board games and casinos, typically weight 4.1 g. 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 Battery (AAA).
In other words, 5.6700 grams is 0.470 times the weight of a Battery (AAA), and the weight of a Battery (AAA) is 2.10 times that amount.
(Alkaline)
An "AAA" size battery weighs 12 g. 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 two-and-a-half times as heavy as a Penny.
In other words, the weight of a Penny is 0.430 times 5.6700 grams.
(United States) (formally one-cent coin) (Union shield design, 2010-present)
The weight of a United States penny is 2.5 g. 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.
It's about two-fifths as heavy as a Compact disc.
In other words, 5.6700 grams is 0.380 times the weight of a Compact disc, and the weight of a Compact disc is 2.60 times that amount.
(a.k.a. CD)
A compact disc typically weighs 15 g. 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.