Q. A new system of units is proposed in which unit of mass is $\alpha\, kg$, unit of length is $\beta\, m$ and unit of time is $\gamma\, s$. What will be value of $5 \,J$ in this new system ?
Physical World, Units and Measurements
Solution:
Joule is a unit of energy.
$SI\quad$
New system
$n_1=5\,\,$
$n_2=?$
$M_1=1\,kg\,\,$
$M_2=\alpha \,kg$
$L_1=1\,m\,\,$
$L_2 =\beta \,m$
$T_1 =1\,s\,\,$
$T_2\,\gamma \,s$
Dimensional formula of energy is $[ML^2T^{-2}]$. Comparing with $[ML^2T^{-2}]$, we get
$a = 1$, $b = 2$, $c = -2$
As $n_{2}=n_{1}\left(\frac{M_{1}}{M_{2}}\right)^{a}\left(\frac{L_{1}}{L_{2}}\right)^{b}\left(\frac{T_{1}}{T_{2}}\right)^{c}$
$=5\left(\frac{1\,kg}{\alpha\,kg}\right)^{1}\left(\frac{1\,m}{\beta\,m}\right)^{2}\left(\frac{1\,s}{\gamma\,s}\right)^{-2}$
$=\frac{5\gamma^{2}}{\alpha\beta^{2}}=5\alpha^{-1}\beta^{-2}\gamma^{2}$
| $SI\quad$ | New system |
|---|---|
| $n_1=5\,\,$ | $n_2=?$ |
| $M_1=1\,kg\,\,$ | $M_2=\alpha \,kg$ |
| $L_1=1\,m\,\,$ | $L_2 =\beta \,m$ |
| $T_1 =1\,s\,\,$ | $T_2\,\gamma \,s$ |