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In the expression P =E I^2 m^(-5) G^(-2)...

In the expression `P =E I^2 m^(-5) G^(-2),` E, m, I and G denote energy, mass, angular momentum and gravitational constant, respectively. Show that P is a dimensionless quantity.

Text Solution

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`"Given, expression is "P=EL^(2)m^(-5)G^(-2)`
`"where E is energy "[E]=[ML^(2)T^(-2)]`
`"m is mass "[m]=[M]`
`"L is angular momentum "[L]=[ML^(2)T^(-1)]`
`"G is gravitational constant "[G]=[M^(-1)L^(3)T^(-2)]`
Substituting dimensions of each term in the given expression,
`[P]=[ML^(2)T^(-2)]xx[ML^(2)T^(_1)]^(2)xx[M]^(-5)xx[M^(-1)L^(3)T^(-2)]^(-2)`
`=[M^(1+2-5+2)L^(2+4-6)T^(-2-2+4)]=[M^(0)L^(0)T^(0)]`
Therefore, P is a dimensionless quantity.
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