Q1.6 The U.S. National Institute of Science and Technology (NIST) maintains several accurate copies of the international standard kilogram. Even after careful cleaning, these national standard kilograms are gaining mass at an average rate of about 1μg/y (1 y = 1 year) when compared every 10 years or so to the standard international kilogram. Does this apparent change have any importance? Explain.
Response: The kilogram is the only fundamental constant not based on a universal constant. The second (time) is defined as the time required for 9,129,631,770 cycles microwave radiation of a cesium atom, and the meter (distance) is based on the distance that light travels in a vacuum in 1/299,792,458 of a second. If the mass of the kilogram standard changes, then the mass of actual kilogram changes as well. There is a Veritasium video about fixing the mass of a kilogram to a new standard involving a sphere of a specific isotope of silicon.
Q1.7 What physical phenomena (other than a pendulum or a cesium clock) could you use to define a time standard?
Response: We need a very regular periodic natural phenomena. A day is very regular (although it does change over a long timescale). Sundials are among the first types of clocks developed by humans. A moon cycle as seen from Earth is called a lunation and the cycle takes exactly 29.5305882 days. People have also used water clocks and hourglasses to keep track of time (although these are never quite precise) and would not make for good standards. On a shorter timescale, the best bet is probably a pulsar. A pulsar is a highly magnetized neutron star that emits a beam of electromagnetic radiation on a very regular interval. In fact a pulsar clock has been built and uses radio signals emitted from pulsars to keep time.
Q1.8 Describe how you could measure the thickness of a sheet of paper with an ordinary ruler.
Response: I could stack x sheets of paper until the thickness of the stack measured one centimeter. Then, the thickness of one sheet of paper would be approximately equal to (1/x) centimeters. A quick google search says that a sheet of paper is about 0.1mm thick, so 10 sheets would equal 1mm and 100 sheets would equal 1cm.
Q1.9 The quantity π = 3.14159... is a number with no dimensions since it is the ratio of two lengths. Describe two or three other geometrical or physical quantities that are dimensionless.
Response: √2 = 1.41421... This is the ratio of the hypotenuse of an isosceles right triangle to one of the legs.
(√3)/2 = 0.86602... This is the ratio of an altitude of an equilateral triangle to one of its sides.
Φ = 1.618033... This is the ratio of the diagonal of a regular pentagon to one of its sides.
Q1.10 What are the units of volume? Suppose another student tells you that a cylinder of radius r and height h has volume given by πr3h. Explain why this cannot be right.
Response: Any unit of length gives a corresponding unit of volume, namely the volume of a cube whose side has the given length. For example, a cubic centimetre (cm3) would be the volume of a cube whose sides are one centimetre (1 cm) in length. In the International System of Units (SI), the standard unit of volume is the cubic metre (m3). The metric system also includes the litre (L) as a unit of volume, where one litre is the volume of a 10-centimetre cube. Thus 1 litre = (10 cm)3 = 1000 cubic centimetres = 0.001 cubic metres, so 1 cubic metre = 1000 litres.
The expression πr3h yields a quantity with units of length raised to the 4th power, and thus cannot represent a volume (example: r = 1m, h = 2m, then πr3h = π(1m)(1m)(1m)(2m) = 2π m4)
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