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How heavy is 0.001 ounces?

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It's about one-one-thousand-five-hundredth as heavy as a Golf Ball.
In other words, 0.001 ounces is 0.0006172 times the weight of a Golf Ball, and the weight of a Golf Ball is 1,620 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 1.620 ounces. 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 one-two-thousandth as heavy as a Tennis Ball.
In other words, 0.001 ounces is 0.000491 times the weight of a Tennis Ball, and the weight of a Tennis Ball is 2,040 times that amount.
(ITF Standard; for Type 2 ball)
Per International Tennis Federation standard, regulation tennis balls must weigh about 2.04 ounces. 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-two-thousand-five-hundredth as heavy as a Battery (C).
In other words, the weight of a Battery (C) is 2,300 times 0.001 ounces.
(Alkaline)
A "C" size battery weighs 2.30 ounces. C batteries are typically used in medium-drain devices such as musical instruments.
It's about one-three-thousand-five-hundredth as heavy as a Deck of Cards.
In other words, the weight of a Deck of Cards is 3,300 times 0.001 ounces.
(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 3.30 ounces. 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 8,000 ounces.
It's about one-five-thousandth as heavy as a Battery (D).
In other words, 0.001 ounces is 0.00021 times the weight of a Battery (D), and the weight of a Battery (D) is 4,760 times that amount.
(Alkaline)
A "D" size battery weighs 4.760 ounces. 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-five-thousandth as heavy as a Baseball.
In other words, 0.001 ounces is 0.00019512195121951 times the weight of a Baseball, and the weight of a Baseball is 5,125.0000000001 times that amount.
(MLB rules)
Per Major League Baseball rules, a baseball must weigh between 5 ounces and 5.250 ounces. A major league pitcher can throw a baseball at up to 150 kph (91 mph) (with a fastball pitch), with record-setting pitches as fast as 171 kph (106 mph).
It's about one-six-thousandth as heavy as a Billiard Ball.
In other words, 0.001 ounces is 0.00017391304347826 times the weight of a Billiard Ball, and the weight of a Billiard Ball is 5,750 times that amount.
(American-style, WPA specifications)
Per the World Pool-Billiard Association, a billiard ball weight must weigh between 5.50 ounces and 6 ounces. According to legend, the difficulty in obtaining elephant ivory for the manufacture of billiard balls in the 19th century inspired the search for a replacement material, which led to the development of the first industrial plastic.