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] $$=$$ [Xα]; [speed] $$=$$ [Xβ]; [acceleration] $$=$$[Xp]; [linear momentum] $$=$$ [Xq]; [force] $$=$$ [Xr]. Then $$-$$

Question

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] $$=$$ [Xα]; [speed] $$=$$ [Xβ]; [acceleration] $$=$$[Xp]; [linear momentum] $$=$$ [Xq]; [force] $$=$$ [Xr]. Then $$-$$

Infinity Learn · Accepted Answer

Given $$L=x$$α ……(1)LT$$−$$1=x$$β ……$$(2)LT$$−$$2=xp$$ ……$$(3)MLT$$−$$1=xq$$ ……$$(4)MLT$$−$$2=xr$$ ……$$(5)(1)(2)⇒$$T=x$$α−βFrom (3)x$$α$$x2($$α−β$$)=xp$$⇒α$$+p=2$$β From $$(4) $$M=xq$$−β  From (5)⇒$$xq=xrx$$α−β ⇒α$$+r$$−$$q=$$β$$----(6)Replacing$$ value 'α' in equation (6) from (A)2$$β−$$p+r$$−$$q=$$β⇒$$p+q$$−$$r=$$β     $$(B)Replacing$$ value of 'β' in equation $$(6) from $$(A)2$$α$$+2r$$−$$2q=$$α$$+p$$α$$=p+2q$$−$$2r$$

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