Question

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?

• A. $5\alpha\beta^{2}\gamma^{- 2}$
• B. $5\alpha^{- 1}\beta^{- 2}\gamma^{2}$
• C. $5\alpha^{- 2}\beta^{- 1}\gamma^{- 2}$
• D. $5\alpha^{- 1}\beta^{2}\gamma^{- 2}$

Answer 1

Answer: (B)

$5\alpha^{- 1}\beta^{- 2}\gamma^{2}$

Answer 2

Joule is a units of energy

SI New system

$n_{1} = 5$ $n_{2} = ?$

$M_{1} = 1\lg$ $M_{2} = \alpha kg$

$L_{1} = 1m$ $L_{2} = \beta m$

$T_{1} = 1s$ $T_{2} = \gamma s$

Dimensional formula of energy is $\lbrack ML^{2}T^{- 2}\rbrack$. Comparing with $\lbrack M^{a}L^{b}T^{c}\rbrack,weget$

As$n_{2} = n_{1}\left( \frac{M_{1}}{M_{2}} \right)^{a}\left( \frac{L_{1}}{L_{2}} \right)^{b}\left( \frac{T_{1}}{T_{2}} \right)^{c}$

$= 5\left( \frac{1kg}{\alpha kg} \right)^{1}\left( \frac{1m}{\beta m} \right)^{2}\left( \frac{1s}{\gamma s} \right)^{- 2} = \frac{5\gamma^{2}}{\alpha\beta^{2}} = 5\alpha^{- 1}\beta^{- 2}\gamma^{2}$