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How heavy is 0.001 ounces?

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It's about one-fiftieth as heavy as a Paper Clip.
In other words, the weight of a Paper Clip is 50 times 0.001 ounces.
(average)
Notwithstanding the tremendous variation in sizes and materials, a typical paperclip weighs about 0.049 ounces. The country of Norway has used the paper clip as something of a national symbol since the end of World War II based on the erroneous notion that the paperclip was invented by a Norwegian, Johan Vaaler. However, the paperclip had already been in existence for at least 30 years by the time of Vaaler patented his paperclip design.
It's about one-sixty-fifth as heavy as a Playing Card.
In other words, the weight of a Playing Card is 63 times 0.001 ounces.
(ISO 216 B8 size, 2.5" x 3.5")
A single ISO 216 B8-size playing card weighs 0.063 ounces. Although this size is the most common home playing card, many casinos use "bridge size" cards — 2,640 sq. mm (4.036 sq. in) smaller in area.
It's about one-ninetieth as heavy as a Penny.
In other words, the weight of a Penny is 91 times 0.001 ounces.
(United States) (formally one-cent coin) (Union shield design, 2010-present)
The weight of a United States penny is 0.088 ounces. According to a 2010 report by the United States Mint, the cost of manufacturing and distributing a penny is $0.0179 — more than its face-value.
It's about 95 times as heavy as a Poppy Seed.
In other words, the weight of a Poppy Seed is 0.011 times 0.001 ounces.
(Papaver somniferum)
Fifty percent oil by weight, a poppy seed weighs approximately 0.0000110 ounces. The true pigment of a poppy seed is brown in color, while the characteristic bluish color is an optical illusion caused by a layer of calcium oxalate crystals refracting light rays.
It's about one-one-hundred-fiftieth as heavy as a Die (Dice).
In other words, the weight of a Die (Dice) is 140 times 0.001 ounces.
(for six-sided, 16 mm, rounded cube)
A die, of the size most commonly used in board games and casinos, typically weight 0.140 ounces. 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.
It's about one-one-hundred-fiftieth as heavy as a sheet of Paper.
In other words, the weight of a sheet of Paper is 200 times 0.001 ounces.
(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 300 ounces, 8.5 inch x 11 inch paper is 70 ounces, each sheet weighing 0.20 ounces. Paper of this size has a thickness of 0.004 caliper (0.1 mm).
It's about one-two-hundredth as heavy as a Quarter.
In other words, the weight of a Quarter is 200 times 0.001 ounces.
(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 0.20 ounces each. In answer to the common riddle, there are 119 ridges around the edge of the coin in its current design.