Theory of Dimensionless Products: General method for transforming coordinate of a magnitude

General method for transforming coordinate of a magnitude

Coordinate of a point

The coordinate of an arbitrary is defined as

The number that specifies the distance from an origin on an axis to the point P.

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

coordinate of the magnitude

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Magnitude (i.e coordinate of magnitude) change when dimensional unit changes.

When the unit of a dimension is changed, the magnitude (coordinate of magnitude) will change accordingly. For example, change of units for length dimension from meter to centimeter results in appropriate change in the coordinate of magnitude of the same point P. Here, a general method for transforming the coordinate of a magnitude will be given. ❷

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".

Coordinate of the magnitude for M, L and T

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.

The question then is

What is the relationship between x and x̄?

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Solution: Algebraic formula for transformation

Let us use the notation

units	Original	New
mass	OM	NM
length	OL	NL
time	OT	NT

Then

1 OM = A ⋅ NM
1 OL = B ⋅ NL
1 OT = C ⋅ NT

and x[(OM)^a ⋅ (OL)^b ⋅ (OT)^c] = x[A^a(NM)^a ⋅ B^b(NL)^b ⋅ C^c(NT)^c]

Multiplying both sides by [(NM)^a ⋅ (NL)^b ⋅ (NT)^c]⁻¹ we get,

x[(OM)^a ⋅ (OL)^b ⋅ (OT)^c] ⋅ [(NM)^a ⋅ (NL)^b ⋅ (NT)^c]⁻¹ = x[A^a ⋅ B^b ⋅ C^c].

Let, x̄ ≜ x[(OM)^a ⋅ (OL)^b ⋅ (OT)^c] ⋅ [(NM)^a ⋅ (NL)^b ⋅ (NT)^c]⁻¹ and substituting A^a = (OM/NM)^a, B^b = (OL/NL)^b and B^c= (OT/NT)^c, we get x̄ = x ⋅ A^a ⋅ B^b ⋅ C^c is the algebraic formula for transformation.

Generalizing this for the seven fundamental dimensions [M], [L], [T], [A], [K], [cd], and [mol] we have

x̄ = x ⋅ A^a ⋅ B^b ⋅ C^c ⋅ D^d ⋅ E^e ⋅ F^f ⋅ G^g

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Eg.1: Given, 900 ft/min² what is the coordinate of magnitude for the changed units meter/sec²?

Since, the original unit is ft/min²

(OM)^a ⋅ (OL)^b ⋅ (OT)^c → (OM)⁰ ⋅ ft¹ ⋅ min⁻² Hence, a = 0, b = 1, c = −2, and the coordinate of magnitude for the original unit, x = 900.

For the new unit meter/sec² we know that

1 ft = 0.3048 ⋅ meter
1 min = 60 ⋅ sec

Hence, B = 0.3048 and C = 60.

To find the coordinate of magnitude for the changed unit

x̄ = x ⋅ A^a ⋅ B^b ⋅ C^c = 900 ⋅ A⁰ ⋅ 0.3048¹ ⋅ 60⁻² = 0.0762 Therefore, the coordinate of magnitude with the changed unit is 0.0762 meter/sec².

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Next:

Dimensional Homogeneity (p:2) ➽

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