First slide

Dimensions of physical quantities

Question

In SI system, the magnitude work done is joule. In another system, where the fundamental physical quantities are in gram, centimetre and second, the
magnitude of work is

Moderate

A

$10^{7} e r g$
B

$10^{5} e r g$
C

$10^{8} e r g$
D

$10^{10} e r g$

Solution

Joule is SI unit of work. The dimensions of work in SI.
$= [W_{1}] = [M_{1} L_{1}^{2} T_{1}^{- 2}]$
But erg is CGS unit of work. The dimensions of work in
CGS unit $= [W_{2}] = [M_{2} L_{2}^{2} T_{2}^{- 2}]$
As, $n_{1} (u_{1}) = n_{2} (u_{2})$
or $n_{1} [M_{1} L_{1}^{2} T_{1}^{- 2}] = n_{2} [M_{2} L_{2}^{2} T_{2}^{- 2}]$
Here, $M_{1} = k g, L_{1} = m, T_{1} = s$
$M_{2} = g, L_{2} = c m, T_{2} = s$
$M_{1} = 1000 M_{2},$
$L_{1} = 100 L_{2}$
$T_{1} = T_{2} a n d n = 1$
$∴ n_{2} = n_{1} [\frac{M_{1}}{M_{2}}] {[\frac{L_{1}}{L_{2}}]}^{2} {[\frac{T_{1}}{T_{2}}]}^{- 2}$
$= 1 [\frac{1000 M_{2}}{M_{2}}] {[\frac{100 L_{2}}{L_{2}}]}^{2} {[\frac{T_{1}}{T_{2}}]}^{2} = 10^{7}$
$∴ 1 j o u l e = 10^{7} e r g$

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