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] =[X r ]. 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] =[X r ]. Then (A) α+ p =2 β (B) p + q - r =β (C) p - q + r =α (D) p + q + r =β

Tardigrade Admin · Accepted Answer

r=xα v=xβ a=xp P=xq F=xr (A) ra arrow[ x α+ P ] [( text Energy / text mass )]=[2 β] (B) [( aP / F )]=[ at ]=[ v ]=β P + q - r =β

$α + p = 2 β$

$p + q - r = β$

$p - q + r = α$

$p + q + r = β$

$r = x^{α}$
$v = x^{β}$
$a = x^{p}$
$P = x^{q}$
$F = x^{r}$
(A) $r a \to [x^{α + P}]$
$[\frac{Energy}{mass}] = [2 β]$
(B) $[\frac{a P}{F}] = [a t] = [v] = β$
$P + q - r = β$

α+p=2β

p+q−r=β

p−q+r=α

p+q+r=β

r=xα v=xβ a=xp P=xq F=xr (A) ra→[xα+P] [ mass Energy ​]=[2β] (B) [FaP​]=[at]=[v]=β P+q−r=β

$α + p = 2 β$

$p + q - r = β$

$p - q + r = α$

$p + q + r = β$

$r = x^{α}$
$v = x^{β}$
$a = x^{p}$
$P = x^{q}$
$F = x^{r}$
(A) $r a \to [x^{α + P}]$
$[\frac{Energy}{mass}] = [2 β]$
(B) $[\frac{a P}{F}] = [a t] = [v] = β$
$P + q - r = β$