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How heavy is 370 grains?

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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 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 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 two-fifths as heavy as a Tennis Ball
In other words, 370 grains is 0.4160 times the weight of a Tennis Ball, and the weight of a Tennis Ball is 2.40 times that amount.
(ITF Standard; for Type 2 ball)
Per International Tennis Federation standard, regulation tennis balls must weigh about 890 grains. Competition balls are also tested for bounce height, with the standards requiring that a ball dropped from 254 cm must bounce back to a height of 135 cm to 147 cm.
It's about half as heavy as a Golf Ball
In other words, 370 grains is 0.5220 times the weight of a Golf Ball, and the weight of a Golf Ball is 1.9160 times that amount.
(per R&A USGA Rules of Golf; maximum rule weight)
According to the Rules of Golf (as approved by the United States Golf Association and the Rules Committee of the Royal and Ancient Golf Club of Saint Andrews), golf balls must weigh no more than 708.80 grains. Golf balls are not required to have the familiar dimpled pattern, but the design has been popular since its invention in 1905 because it reduces drag while increasing lift.
It's about seven-tenths as heavy as a Light Bulb
In other words, 370 grains is 0.710 times the weight of a Light Bulb, and the weight of a Light Bulb is 1.40 times that amount.
(a.k.a. lamp) (for A19-size, 60w, incandescent bulb by Bulbrite; package weight)
A standard, sixty-watt light bulb weighs about 530 grains. The tightly-coiled filament inside a bulb of this size would measure about 579mm if uncoiled and stretched out.
It's about as heavy as a Battery (AA)
(Alkaline)
An "AA" size battery weighs 370 grains. AA batteries came into common use during World War II and were given their name by the American National Standards Institute (ANSI) Committee C18 because they were smaller than the "A" size batteries which were invented first.