Question

A new system of units is proposed in which unit of mass is a kg unit of lenth $\beta$ m and unit of time y s . How much will 5 J measure in this new system ?

Answer 1

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 , lenghth and time in given two sysytems .

So, $n_2=n_1{\left[\frac{M_1}{M_2}\right]}^a\times {\left[\frac{L_1}{L_2}\right]}^b\times {\left[\frac{T_1}{T_2}\right]}^c$ ...................(i)

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

$n_2$ = New sysytem of unit = ? , $n_1$
S.I system of unit = 5 J
$M_2=akg{M}_1=1kg$
$L_2=\beta m,L_1=1m$
$T_2=ys,T_1=1s$
Therefore by putting the values in eqaution (i) we get

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

$n_2=5\left[\frac{1}{\alpha }\right]\times {\left[\frac{1}{\beta }\right]}^2\times {\left[\frac{1}{y}\right]}^{-2}$

=$n_{2}5\frac{1}{a}\times \frac{1}{{\beta }^2}\times \frac{1}{y^{-2}}$

=$n_2\frac{5y^2}{\alpha {\beta }^2}$ J

Required value of energy in new system of units

Final answer is =$n_2\frac{5y^2}{a{\beta }^2}$ J