A new system of units is proposed in which unit of mass is \alpha kg, unit of length \beta m and unit of time γ s. How much will 5 J measure in this new system?

Answers (1)

Joules is a unit of work/energy, so let us first write the dimensions of energy-

Energy =[ML^{2}T^{-2}]

Let n_{1} be the SI system of unit & n_{2} be the new system of unit

Thus, n_{2}u_{2}=n_{1}u_{1}

n_{2}=n_{1}(\frac{u_{1}}{u_{2}})

=n_{1}[\frac{M_{1}}{M}_{2}]^{1}[\frac{L_{1}}{L_{2}}]^{2}[\frac{T_{1}}{T_{2}}]^{-2}

Now,

M_{1}=1\; kg                 & M_{2}=\alpha \; kg

L_{1}= 1 \; m                   & L_{2}= \beta \; m

T_{1}= 1 \; second         & T_{2}= \gamma \; sec

Thus, n_{2}= 5[\frac{1\; kg}{\alpha \; kg}]^{1}[\frac{1\; m}{\beta \; m}]^{2}[\frac{1\; s}{\gamma \; s}]^{-2}

              =5[\alpha ^{-1}\beta ^{-2}\gamma ^{2}]

Thus, by the new system,     [\alpha ^{-1}\beta ^{-2}\gamma ^{2}].

 

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