Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are giv

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Sometimes it is convenient to construct a system of units so that all quantities can be expressed in terms of only one physical quantity. In one such system, dimensions of different quantities are given in terms of a quantity X as follows: position = X α ; speed = X β ; acceleration = X p ; linear momentum = X q ; force = X r . Then

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momentum Force = X p - r ⇒ m v m a = X p - r ⇒ v a = X p - r but given that v a = X β - p So, p - r = β - p ⇒ 2 p = β + r . . . 1 s p e e d 2 a c c e l e r a t i o n = p o s i t i o n ⇒ X 2 β X p = X α ⇒ 2 β - p = α ⇒ p + α = 2 β . . . 2 and f o r c e l i n e a r m o m e n t u m = a c c e l e r a t i o n s p e e d ⇒ X r X q = X p X β ⇒ r - q = p - β ⇒ p + q - r = β . . . 3

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