Name: Limitation of Dimensional Analysis
Uploaded: 2019-07-19
Duration: 5 min 9 s
Description: Dimensional analysis can derive the relation between two physical parameters. But does it work in all the cases? Is it possible? Let us discuss this in this video.

Question

Convert

1 J

to

ergs

.

Accepted Answer

The SI unit of energy is given by, $joule (J)$ ,
CGS unit of energy is given by, $erg$ .

The dimensional formula of any derived quantity from mechanics is given by, $M^{a} L^{b} T^{c}$ , where $a, b and c$ represent dimensions $M, L and T$ .

From the definition of work, we have,

$[Work] = [force] \times [distance] = [mass] \times [acceleration] \times [length]$
$\Rightarrow [Work] = [mass] \times \frac{[length]}{{[time]}^{2}} \times [length]$ .

From the dimension of acceleration, we have, $[acceleration] = \frac{[length]}{[time]^{2}}$ .

Dimensional formula of work $= [W] = M^{1} L^{2} T^{- 2}$ , so, dimensions of $M, L and T$ are $a = 1, b = 2, c = - 2$ .

Now,

SI system CGS system

$M_{1} = 1 kg$ $M_{2} = 1 g$

$L_{1} = 1 m$ $L_{2} = 1 cm$

$T_{1} = 1 s$ $T_{2} = 1 s$

From the principle of dimensional homogeneity, we have, $n_{1} {[M_{1}]}^{a} {[L_{1}]}^{b} {[T_{1}]}^{c} = n_{2} {[M_{2}]}^{a} {[L_{2}]}^{b} {[T_{2}]}^{c}$
$\Rightarrow n_{2} = n_{1} {[\frac{M_{1}}{M_{2}}]}^{a} {[\frac{L_{1}}{L_{2}}]}^{b} {[\frac{T_{1}}{T_{2}}]}^{c}$ .

For our case, $1 J = n_{2} erg$ , so $n_{1} = 1$ ,
$\Rightarrow n_{2} = 1 {[\frac{1 kg}{1 g}]}^{1} {[\frac{1 m}{1 cm}]}^{2} {[\frac{1 s}{1 s}]}^{- 2}$
$\Rightarrow n_{2} = 1 {[\frac{1000 g}{1 g}]}^{1} {[\frac{100 cm}{1 cm}]}^{2} = 10^{7} (∵ 1 kg = 1000 g, 1 m = 100 cm)$

$\Rightarrow n_{2} = 10^{7}$ .

So $1 J = 10^{7} erg$ .

Convert $1 J$ to $ergs$ .

Important Questions on Dimensions and Measurement

Learn and Prepare for any exam you want !

Convert 1 J to ergs.

Important Questions on Dimensions and Measurement

Learn and Prepare for any exam you want !

Convert $1 J$ to $ergs$ .