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How heavy is 370 grains?

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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.
It's about one-and-one-tenth times as heavy as a Mouse.
In other words, the weight of a Mouse is 0.910 times 370 grains.
(a.k.a. "House mouse", Mus musculus) (adult)
The weight of a mouse is typically an average of 320 grains. While most mice typically live two to three years, the Methuselah Mouse Prize — an award administered by a non-profit organization which advances aging-related research — was once awarded to a team of scientists whose research resulted in a mouse living just under five years.
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 one-and-three-fifths times as heavy as a Compact disc.
In other words, the weight of a Compact disc is 0.630 times 370 grains.
(a.k.a. CD)
A compact disc typically weighs 230 grains. 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 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 two times as heavy as a Battery (AAA).
In other words, the weight of a Battery (AAA) is 0.50 times 370 grains.
(Alkaline)
An "AAA" size battery weighs 190 grains. 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-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.