Ncert Exercises & Solutions

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

Hint: Use the principle of conversion of units.

Step 1: Find the dimensions of energy.

$\begin{matrix} We know that dimension of energy = [{ML}^{2} T^{- 2}] Let M_{1}, L_{1}, T_{1} and M_{2}, L_{2}, T_{2} are units of mass, length, and time in given two systems . \end{matrix}$ $\begin{array}{l} We know that dimension of energy = [{ML}^{2} T^{- 2}] \\ Let M_{1}, L_{1}, T_{1} and M_{2}, L_{2}, T_{2} are units of mass, length, and time in given two systems . \end{array}$

$\begin{array}{l} ∴ M_{1} = 1 kg, L_{1} = 1 m, T_{1} = 1 \sec \\ M_{2} = α kg, L_{2} = β m, T_{2} = γ \sec \end{array}$

Step 2: Use the conversion of units.

The magnitude of a physical quantity remains the same, whatever be the system of units of its measurement i.e., ${nu}_{1} = n_{2} u_{2}$

$\Rightarrow n_{2} = n_{1} \frac{u_{1}}{u_{2}} = n_{1} \frac{[M_{1} L_{1}^{2} T_{1}^{- 2}]}{[M_{2} L_{2}^{2} T_{2}^{- 2}]} = 5 [\frac{M_{1}}{M_{2}}] \times {[\frac{L_{1}}{L_{2}}]}^{2} \times {[\frac{T_{1}}{T_{2}}]}^{- 2}$
$= 5 [\frac{1}{α} kg] \times [\frac{1}{β} m]^{2} \times [\frac{1}{γ} s]^{- 2}$
$= 5 \times \frac{1}{α} \times \frac{1}{β^{2}} \times \frac{1}{γ^{- 2}}$
$n_{2} = \frac{5 γ^{2}}{{αβ}^{2}}$

Thus, a new unit of energy will be $\frac{γ^{2}}{{αβ}^{2}}$ .

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