Question

The velocity, acceleration and force in two systems of units are related as under:
(i) $v^{\prime }=\frac{{\alpha }^2}{\beta }v$
(ii) $a^{\prime }=\left(\alpha \beta \right)a$
(iii) $F^{\prime }=\left(\frac{1}{\alpha \beta }\right)F$
All the primed symbols belong to one system and unprimed ones belong to the other system. $\alpha$ and $\beta$ are dimensionless constants. Which of the following is/are correct?

• A. Length standards of the two systems are related by:
  $L^{\prime }=\left(\frac{{\alpha }^3}{{\beta }^3}\right)L$
• B. Mass standards of the two systems are related by:
  $m^{\prime }=\left(\frac{1}{{\alpha }^2{\beta }^2}\right)m$
• C. Time standards of the two systems are related by:
  $T^{\prime }=\left(\frac{\alpha }{{\beta }^2}\right)T$.
• D. Momentum standards of the two systems are related by:
  $P^{\prime }=\left(\frac{1}{{\beta }^3}\right)P$

Answer 1

Answer: (A), (B), (C), (D)

Momentum standards of the two systems are related by:

$P^{\prime }=\left(\frac{1}{{\beta }^3}\right)P$
Velocity = length/time, acc=length/${\text{(time)}}^2$
$\Rightarrow$ Length $=\frac{{\text{(velocity)}}^2}{\text{acc.}}$, i.e., $L^{\prime }=\frac{v^2}{a^{\prime }}$ and $L=\frac{v^2}{a}$
$\Rightarrow \frac{L^{\prime }}{L}={\left(\frac{v^{\prime }}{v}\right)}^2\left(\frac{a}{a^{\prime }}\right)={\left(\frac{{\alpha }^2}{\beta }\right)}^2\frac{1}{\alpha \beta }=\frac{{\alpha }^3}{{\beta }^3}$
Now, $m^{\prime }=\frac{F^{\prime }}{a^{\prime }}$ and $m=\frac{F}{a}$
$\Rightarrow \frac{m^{\prime }}{m}=\frac{F^{\prime }}{F}\frac{a}{a^{\prime }}=\frac{1}{\alpha \beta }\times \frac{1}{\alpha \beta }=\frac{1}{{\alpha }^2{\beta }^2}$
Time = vel./acc, i.e., $T^{\prime }=\frac{v^{\prime }}{a^{\prime }}$ and $T=\frac{v}{a}$
$\Rightarrow \frac{T^{\prime }}{T}=\frac{v^{\prime }}{v}\frac{a}{a^{\prime }}=\frac{{\alpha }^2}{\beta }\frac{1}{\alpha \beta }=\frac{\alpha }{{\beta }^2}$
Momentum = mass $\times$ velocity, i.e.,
$P^{\prime }=m^{\prime }{v}^{\prime }$, and $P=mv$
$\Rightarrow \frac{P^{\prime }}{P}=\frac{m^{\prime }}{m}\frac{v^{\prime }}{v}=\frac{1}{{\alpha }^2{\beta }^2}\frac{{\alpha }^2}{\beta }=\frac{1}{{\beta }^3}$