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 $]=\left[X^{\alpha}\right] ;[$ Speed $]=\left[X^{\beta}\right] ;[$ acceleration $]=\left[X^{p}\right] ;[$ Linear momentum $]=\left[X^{q}\right] ;$ [force] $=\left[X^{ r }\right]$. Then

• A. $\alpha+ p =2 \beta$
• B. $p + q - r =\beta$
• C. $p - q + r =\alpha$
• D. $p + q + r =\beta$

Answer 1

Answer: (A), (B)

$r=x^{\alpha} $
$v=x^{\beta} $
$a=x^{p} $
$P=x^{q} $
$F=x^{r}$
(A) $ra \rightarrow\left[ x ^{\alpha+ P }\right]$
$\left[\frac{\text { Energy }}{\text { mass }}\right]=[2 \beta]$
(B) $\left[\frac{ aP }{ F }\right]=[ at ]=[ v ]=\beta$
$P + q - r =\beta$