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How heavy is 0.001 ounces?

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It's about one-thirtieth as heavy as a Nail.
In other words, the weight of a Nail is 28 times 0.001 ounces.
(for 15-gauge, 2-inch, finishing nails)
A two-inch finishing nail weighs 0.028 ounces. Nails have been in use for more than 5,400 years and were ven used as a medium of exchange in ancient times.
It's about one-fortieth as heavy as a Jelly Bean.
In other words, the weight of a Jelly Bean is 38 times 0.001 ounces.
(for Jelly Belly beans; approximate)
Made of sugar, corn syrup, and pectin, a single jelly bean weighs about 0.04 oz. President Ronald Reagan cited jelly beans as his favorite candy and 110,000 ounces of the candies were served at his first inauguration in 1981.
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 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).