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Question: If the speed of light \((c)\), acceleration due to gravity \((g)\) and pressure \((p)\) are taken as...

If the speed of light (c)(c), acceleration due to gravity (g)(g) and pressure (p)(p) are taken as the fundamental quantities, then the dimension of gravitational constant is

A

(a) c2g0p−2c^{2}g^{0}p^{- 2}

A

(b) c0g2p−1c^{0}g^{2}p^{- 1}

A

(c) cg3p−2cg^{3}p^{- 2}

A

(d) c−1g0p−1c^{- 1}g^{0}p^{- 1}

Explanation

Solution

(b)

Sol. Let [G]∝cxgypz\lbrack G\rbrack \propto c^{x}g^{y}p^{z}

by substituting the following dimensions :

[G]=[M−1L3T−2],[c]=[LT−1],[g]=[LT−2]\lbrack G\rbrack = \lbrack M^{- 1}L^{3}T^{- 2}\rbrack,\lbrack c\rbrack = \lbrack LT^{- 1}\rbrack,\lbrack g\rbrack = \lbrack LT^{- 2}\rbrack

[p]=[ML−1T−2]\lbrack p\rbrack = \lbrack ML^{- 1}T^{- 2}\rbrack

and by comparing the powers of both sides

we can get x=0,y=2,z=−1x = 0,y = 2,z = - 1

∴\therefore [G]∝c0g2p−1\lbrack G\rbrack \propto c^{0}g^{2}p^{- 1}