39. In the expression P=El2m5G2, E, m, l, and G denote energy, mass, angular momentum, and gravițational constant, respectively. Show that P is a dimensionless quantity.

 
Hint: Dimensions of LHS = Dimensions of RHS
Step 1: Find the dimensions of E, m, l, and G.

Given, expression is P=El2m5G2where E is energy, [E]=[ML2T2]m is mass, [m]=[M]l is angular momentum, l=[ML2T1]G is gravitational constant, G=[M1L3T2]

Step 2: Find the dimensions of P.

Substituting dimensions of each term in the given expression,

[P]=[ML2T2]×[ML2T1]2×[M]5×[M1L3T2]2=[M1+25+2L2+46T22+4]=[M0L0T0]

Therefore, P is a dimensionless quantity.