Coordinate of a point
Extending the notion of coordinates beyond distance
The above definition designates distance to the number, i.e. coordinate of the point on distance. However, if the dimension of the axis is time, the number for the point P will mean coordinate of the point on time. Similarly, for temperature axis it will mean coordinate of the point on temperature. And so on ...Therefore, the magnitude of any scalar quantity in terms of coordinate can be generalized as
Magnitude (i.e coordinate of magnitude) change when dimensional unit changes.
Problem setup and statement
For simplicity consider the three fundamental dimensions [M], [L], and [T] such that,
- [M^{a} L^{b} T^{c}] is the dimension of an entity where a, b, c ∈ ℜ
- x is the coordinate of the magnitude of the entity when the dimensions mass, length, and time are measured in certain units; let us refer to them as "original units".
Imagine introducting new units such that,
- the orginal to new unit relation is
- 1 original mass unit = A new mass units
- 1 original length unit = B new length units
- 1 original time unit = C new time units
- where, A, B, and C are arbitrary positive constants ∈ ℜ^{+}.
- x̄ is the coordinate of the magnitude in terms of the new units.
What is the relationship between x and x̄?
❸Solution: Algebraic formula for transformation
Let us use the notation
units | Original | New |
---|---|---|
mass | OM | NM |
length | OL | NL |
time | OT | NT |
- 1 OM = A ⋅ NM
- 1 OL = B ⋅ NL
- 1 OT = C ⋅ NT
Multiplying both sides by [(NM)^{a} ⋅ (NL)^{b} ⋅ (NT)^{c}]^{−1} we get,
Generalizing this for the seven fundamental dimensions [M], [L], [T], [A], [K], [cd], and [mol] we have
Eg.1: Given, 900 ft/min^{2} what is the coordinate of magnitude for the changed units meter/sec^{2}?
Since, the original unit is ft/min^{2}
For the new unit meter/sec^{2} we know that
- 1 ft = 0.3048 ⋅ meter
- 1 min = 60 ⋅ sec
To find the coordinate of magnitude for the changed unit
Next:
Dimensional Homogeneity (p:2) ➽