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How heavy is 370 grains?

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It's about two-fifths as heavy as a Battery (C)
In other words, 370 grains is 0.370 times the weight of a Battery (C), and the weight of a Battery (C) is 2.70 times that amount.
(Alkaline)
A "C" size battery weighs 1,000 grains. C batteries are typically used in medium-drain devices such as musical instruments.
It's about three-and-a-half times as heavy as an Eye (Human)
In other words, the weight of an Eye (Human) is 0.290 times 370 grains.
(adult)
A fully developed eye (in a person over the age of thirteen), weighs about 110 grains. 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-fourth as heavy as a Deck of Cards
In other words, 370 grains is 0.260 times the weight of a Deck of Cards, and the weight of a Deck of Cards is 3.80 times that amount.
(ISO 216 B8 size, a.k.a. "poker size" 63 x 88 mm, 52-card deck)
A deck of 52 ISO 216 B8-size playing cards weighs 1,500 grains. Utilizing nearly 3,700 decks, the record-holding largest card tower was built in early 2010, stood over 3.5 m (11 ft), and weighed over 3,500,000 grains.
It's about four times as heavy as a Quarter
In other words, 370 grains is 4.2290 times the weight of a Quarter, and the weight of a Quarter is 0.23650 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 87.50 grains each. In answer to the common riddle, there are 119 ridges around the edge of the coin in its current design.
It's about five times as heavy as a sheet of Paper
In other words, 370 grains is 4.80 times the weight of a sheet of Paper, and the weight of a sheet of Paper is 0.210 times that amount.
(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 140,000 grains, 8.5 inch x 11 inch paper is 31,000 grains, each sheet weighing 77 grains. Paper of this size has a thickness of 0.004 caliper (0.1 mm).
It's about one-fifth as heavy as a Battery (D)
In other words, 370 grains is 0.1780 times the weight of a Battery (D), and the weight of a Battery (D) is 5.620 times that amount.
(Alkaline)
A "D" size battery weighs 2,080 grains. The D size battery is the oldest of the modern battery sizes, first invented by the National Carbon Company in 1898.
It's about six times as heavy as a Die (Dice)
In other words, 370 grains is 5.80 times the weight of a Die (Dice), and the weight of a Die (Dice) is 0.170 times that amount.
(for six-sided, 16 mm, rounded cube)
A die, of the size most commonly used in board games and casinos, typically weight 63 grains. 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.