If area (A), velocity (V) and density ( ρ ) are taken as fundamental units, what is the dimensional formula for force?

Question

If area (A), velocity (V) and density ( ρ ) are taken as fundamental units, what is the dimensional formula for force? (A) [ AV 2 ρ ] (B) [ A 2 V ρ] (C) [ AV ρ2] (D) [ AV ρ]

Tardigrade Admin · Accepted Answer

F = A α V β ρ &gamma; or [ MLT -2] =[ L 2]α[ LT -1]β[ ML -3]&gamma; =[ M &gamma; L 2 α+β-3 &gamma; T -β] This gives: &gamma;=1, β=2, 2 α+β-3 &gamma;=1 or α=1 ; F =[ AV 2 ρ]

Q. If area (A), velocity (V) and density ( $ρ$ ) are taken as fundamental units, what is the dimensional formula for force?

$[A V^{2} ρ]$

$[A^{2} V ρ]$

$[A V ρ^{2}]$

$[A V ρ]$

$F = A^{α} V^{β} ρ^{γ}$
or $[M L T^{- 2}] = [L^{2}]^{α} [L T^{- 1}]^{β} [M L^{- 3}]^{γ}$ $= [M^{γ} L^{2 α + β - 3 γ} T^{- β}]$
This gives : $γ = 1, β = 2, 2 α + β - 3 γ = 1$
or $α = 1; F = [A V^{2} ρ]$

Q. If area (A), velocity (V) and density ( ρ ) are taken as fundamental units, what is the dimensional formula for force?

[AV2ρ]

[A2Vρ]

[AVρ2]

[AVρ]