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How heavy is 708.80 grains?

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It's about as heavy as a Golf Ball.
(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 four-fifths as heavy as a Tennis Ball.
In other words, 708.80 grains is 0.7960 times the weight of a Tennis Ball, and the weight of a Tennis Ball is 1.260 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 one-and-two-fifths times as heavy as a Light Bulb.
In other words, the weight of a Light Bulb is 0.710 times 708.80 grains.
(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 seven-tenths as heavy as a Battery (C).
In other words, 708.80 grains is 0.710 times the weight of a Battery (C), and the weight of a Battery (C) is 1.40 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 one-and-nine-tenths times as heavy as a Battery (AA).
In other words, the weight of a Battery (AA) is 0.530 times 708.80 grains.
(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 half as heavy as a Deck of Cards.
In other words, 708.80 grains is 0.490 times the weight of a Deck of Cards, and the weight of a Deck of Cards is 2 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 times as heavy as a Mouse.
In other words, 708.80 grains is 2.20 times the weight of a Mouse, and the weight of a Mouse is 0.450 times that amount.
(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.