Question

In a new system of units energy, density and power are taken as fundamental units, then the dimensional formula of universal gravitational constant G will be

WITH ALL THE STEPS OF THE SOLUTION

Answer 1

hello friend...!!!

⇒ the dimensional formula of gravitational constant : [M⁻¹L³T⁻²]

⇒ the dimensional unit of energy(E) : [ML²T⁻²]

⇒ the dimensional unit of density(D): [ ML⁻³]

⇒ the dimensional unit of power(P) : [ML²T⁻³]

now,

G = $E^{a}$ x $D^{b}$ x $P^{n}$

[M⁻¹L³T⁻²]  = [ML²T⁻²]ᵃ x [ ML⁻³]ᵇ x [ML²T⁻³]ⁿ

⇒ [M⁻¹L³T⁻²] = M⁽ᵃ ⁺ ᵇ ⁺ ⁿ ⁾  x L⁽²ᵃ ⁻³ᵇ ⁺²ⁿ⁾ x T⁽⁻²ᵃ ⁻³ⁿ ⁾

comparing both LHS and RHS , we get ,

a + b + n = -1 

2a -3b +2n = 3 

-2a - 3n = -2

by solving these three linear equations we get ,

a = -2 and b = -1 and n = 2

therefore ,

G = $E^{a}$ x $D^{b}$ x $P^{n}$

implies ,

G =[ E⁻²D⁻¹P²]

G = [ $\frac{P^{2} }{ E^{2} D}$ ]


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hope it helps...!!!

Answer 2

$your \: answer \: is \: \\ \\ G \: = {{ {E}^{ - 2} D}^{ - 1} P \: }^{2} \:$
For explanation see the attached pics


Hope this helps ya ☺