Question

I did the following:

I cut an irregular sheet out of a piece of paper.
I tied the string with a washer at one end to the push pin.
I punched holes in three spot at different "corners" of the paper.
I hung the paper with the push pin from the first hole and let the washer hang down.
I traced the plumb line with a pencil.
I repeated steps 4 and 5 for the other holes.

a) If you cut an irregular shape out of paper and hang it from a thumbtakc it balances around that point it is hung from. If you draw a line straight down the paper from the thumbtack, how is the mass distributed on either side of the line you draw when it is hanging like this?

b) What does the point where the three lines intersect represent? Explain why this method works.

c) Is the third line necessary to find the center of mass? Why or why not?

Answer 1

1)All definitions are ultimately circular in nature since they depend on concepts which must themselves have definitions, a dependence which can not be continued indefinitely without returning to the starting point. To avoid this vicious circle certain concepts must be taken as primitive concepts; terms which are given no definition.^[2] In geometry, it is frequently the case that the concept of line is taken as a primitive.^[3] In those situations where a line is a defined concept, as in coordinate geometry, some other fundamental ideas are taken as primitives. When the line concept is a primitive, the behaviour and properties of lines are dictated by the axioms which they must satisfy.

In a non-axiomatic or simplified axiomatic treatment of geometry, the concept of a primitive notion may be too abstract to be dealt with. In this circumstance it is possible that adescription or mental image of a primitive notion is provided to give a foundation to build the notion on which would formally be based on the (unstated) axioms. Descriptions of this type may be referred to, by some authors, as definitions in this informal style of presentation. These are not true definitions and could not be used in formal proofs of statements. The "definition" of line in Euclid's Elements falls into this category.^[4] Even in the case where a specific geometry is being considered (for example, Euclidean geometry), there is no generally accepted agreement among authors as to what an informal description of a line should be when the subject is not being treated formally.

2)

he notion of line or straight line was introduced by ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects. Until the seventeenth century, lines were defined like this: "The line is the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which [