39. In the expression P=El2m−5G−2P=El2m−5G−2, E, m, l, and G denote energy, mass, angular momentum, and gravițational constant, respectively. Show that P is a dimensionless quantity.
Given, expression is P=El2m−5G−2where E is energy, [E]=[ML2T−2]m is mass, [m]=[M]l is angular momentum, [l]=[ML2T−1]G is gravitational constant, [G]=[M−1L3T−2]
Step 2: Find the dimensions of P.
Substituting dimensions of each term in the given expression,
[P]=[ML2T−2]×[ML2T−1]2×[M]−5×[M−1L3T−2]−2=[M1+2−5+2L2+4−6T−2−2+4]=[M0L0T0]
Therefore, P is a dimensionless quantity.
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