First slide

Dimensions of physical quantities

Question

The value of gravitation constant is $G = 6.67 \times 10^{- 11} N \cdot m^{2} / {kg}^{2}$ in SI units. Convert it into $C G S$ system of units.

Moderate

A

$6.67 \times 10^{- 8} dyne {cm}^{- 2} / g^{2}$
B

$6.67 \times 10^{8} dyne {cm}^{2} / g^{2}$
C

$6.67 \times 10^{- 8} dyne {cm}^{2} / g^{2}$
D

$6.67 \times 10^{- 8} dyne {cm}^{2} / g^{- 2}$

Solution

The dimensional formula of G is $[M^{- 1} L^{3} T^{- 2}]$ .
Using Eq.
$n_{1} [M_{1}^{- 1} L_{1}^{3} T_{1}^{- 2}] = n_{2} [M_{2}^{- 1} L_{2}^{3} T_{2}^{- 2}] n_{2} = n_{1} {[\frac{M_{1}}{M_{2}}]}^{- 1} {[\frac{L_{1}}{L_{2}}]}^{3} {[\frac{T_{1}}{T_{2}}]}^{- 2}$
Here, $n_{1} = 6.67 \times 10^{- 11} M_{1} = 1 kg, M_{2} = 1 g = 10^{- 3} kg, L_{1} = 1 m, L_{2} = 1 cm = 10^{- 2} m, T_{1} = T_{2} = 1 s$
Substituting in the above equation, we get
$n_{2} = 6.67 \times 10^{- 11} {[\frac{1 kg}{10^{- 3} kg}]}^{- 1} {[\frac{1 m}{10^{- 2} m}]}^{3} {[\frac{1 s}{1 s}]}^{- 2}$
or, $n_{2} = 6.67 \times 10^{- 8}$
Thus value of G in CGS system of units is $6.67 \times 10^{- 8} dyne {cm}^{2} / g^{2}$ .

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