First slide

Dimensions of physical quantities

Question

A new system of units is proposed in which unit of mass is $α$ kg, unit of length is $β$ m and unit of time is $γ$ s. What will be value of 5J in this new system?

Moderate

A

$5 α β^{2} γ^{- 2}$
B

$5 α^{- 1} β^{- 2} γ^{2}$
C

$5 α^{- 2} β^{- 1} γ^{- 2}$
D

$5 α^{- 1} β^{2} γ^{2}$

Solution

Joule is a unit of energy.
SI New system
$n_{1} = 5$ $n_{2} = ?$
$M_{1} = 1 kg$ $M_{2} = α kg$
$L_{1} = 1 m$ $L_{2} = β m$
$T_{1} = 1 s$ $T_{2} = γ s$
Dimensional formula of energy is $[{ML}^{2} T^{- 2}]$ . Comparing with $[M^{a} L^{b} T^{c}]$ , we get $a = 1, b = 2, c = - 2$
As $n_{2} = n_{1} {(\frac{M_{1}}{M_{2}})}^{a} {(\frac{L_{1}}{L_{2}})}^{b} {(\frac{T_{1}}{T_{2}})}^{c}$
$=5 {(\frac{1 kg}{α kg})}^{'} {(\frac{1 m}{β m})}^{2} {(\frac{1 s}{γ s})}^{- 2} = \frac{5 γ^{2}}{α β^{2}} = 5 α^{- 1} β^{- 2} γ^{2}$

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