Question

In a new system the unit of mass is $\alpha$kg, unit of length is $\beta$m and unit of time is $\gamma$s. The value of $1J$ in this new system is

• A. ${\gamma }^{2.}/\alpha {\beta }^2$
• B. $\gamma \alpha /bet{a}^2$
• C. $\alpha \beta \gamma$
• D. $\alpha {\gamma }^2/{\beta }^2$

Answer 1

The correct option is A ${\gamma }^{2.}/\alpha {\beta }^2$

Let the physical quantity be $Q=n_1{u}_1=n_2{u}_2$

Let $M_1,L_1,T_1$ and $M_2{L}_2{T}_2$ are units of mass, length and time in given two syatems.

So, $n_2=n_1[\frac{M_1}{M_2}{]}^a\times [\frac{L_1}{L_2}{]}^b\times [\frac{T_1}{T_2}{]}^c$

We know that dimension of energy $\left[U\right]=\left[M{L}^2{T}^{-2}\right]$

According to the problem, $M_1=1kg,L_1=1m,T_1=1s$

$M_2=\alpha kg,L_2=\beta m,T_2=\gamma s$

Substituting the values, we get,

$n_2=\left[\frac{M_1}{M_2}\right]\times [\frac{L_1}{L_2}{]}^2\times [\frac{T_1}{T_2}{]}^{-2}$

$=\left[\frac{1}{\alpha }kg\right]\times [\frac{1}{\beta }m{]}^2\times [\frac{1}{\gamma }s{]}^{-2}$

$=\frac{1}{\alpha }\times \frac{1}{{\beta }^2}\times \frac{1}{{\gamma }^{-2}}=\frac{{\gamma }^2}{\alpha {\beta }^2}J$

This is the required value of energy in the new system of units.