Find the dimension of universal gravitational constant taking the units of density D velocity V and work W as base units instead of those of mass, distance and time.

Find the dimension of universal gravitational constant taking the unit

1.	Find the dimension of universal gravitational constant taking the units of density D velocity V and work W as base units instead of those of mass, distance and time.
Answer» Solution :From Newton.s LAW of gravitation F = `G(M_(1)M_(2))/r^(2)` we [G] = `([F][r^(2)])/[[m^(2)])=(MLT^(-2)xxL^(2))/(M^(2))=M^(-1)L^(3)T^(-2)` Let`M^(-1)L^(3)T^(-2)=D^(X)V^(y)W^(z)` `=(ML^(-3))^(x)xx(LT^(-1))^(y)xx(ML^(2)T^(-2))^(z)` `= M^(x+z)xxL^(-3x+y+2z)XXT^(-y-2z)` Equating powers of same bases, x+z = -1 -3x+y+2z=3 and `""` -y-2z = -2 Solving x,y,z, we get x = `-(1)/(3),y=(10)/(3) "and"z=-(2)/(3)` `:." "[G]="D"^(-(1)/(3))"V"^(10/(3))"W"^(-(2)/(3))`