15  Dimensions and units

Reading questions

\[ \newcommand{\dnorm}{\text{dnorm}} \newcommand{\pnorm}{\text{pnorm}} \newcommand{\recip}{\text{recip}} \]

Reading question 15.1 Give three examples of fundamental dimensions. Provide at least two units that are suitable for measuring each of the dimensions.

question id: fundamental-three

In this book, we denote the dimension of a quantity \(a\) as \([a]\). This is a special use of the square brackets and has nothing to do with arithmetic. (For computer programmers: the dimension interpretation of \([\,]\) is also unrelated to indexing arrays.)

Reading question 15.2  

  1. Suppose that the quantity \(a - b\) makes physical sense. Is it true that $[a-b] = [a] = [b]?

It’s true only if \(a = b\).

It’s obviously false, unless \(b=0\).

It’s true (conditionally on the given supposition that \(a - b\) makes physical sense).

question id: dimension-brackets-1

  1. What does \([a] - [b]\) mean?

It’s the same as \([a - b]\).

It’s zero.

It is without meaning.

It’s the same as \([a]\).

question id: dimension-brackets-2

  1. Does \([a\,b] = [a]\,[b]\)?

Never.

The right hand side is meaningless. You can’t do arithmetic on dimensions.

Sometimes, so long as \([a]\) and \([b]\) are different.

Always, regardless of what \([a]\) and \([b]\) are.

question id: dimension-brackets-3

  1. What does \([17] = [1]\) mean?

It’s nonsense. Obviously \(17 \neq 1\).

Trick question. There’s no such thing as \([17]\).

All pure numbers are dimensionless, so both 17 and 1 have the same null dimension.

question id: dimension-brackets-4

  1. In the quantity \(e^b\), what can you say about the dimension of \(b\), that is, [b]$?

No information is given about \([b]\).

It must be that \([b] = 1\).

\(b\) can have any dimension whatsoever.

question id: dimension-brackets-5

Reading question 15.3  

  1. Give an example of a type of quantity that is dimensionless, yet has units.

question id: flavor-of-1-1

  1. What is the main use of “flavors of one”?

To change the units of a quantity.

To change the dimension of a quantity.

To confuse students.

Quantities don’t have flavors, and 1 is not a flavor either.

question id: flavor-of-1-2

  1. Suppose you have measured an angle to be \(31^\circ\). What is the dimension of the angle?

L

L2

Angle is another fundamental dimension, just as M, L, and T are fundamental dimensions.

Angles are dimensionless.

question id: flavor-of-1-3

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