If force [F], acceleration [A] and time [T] are chosen as the fundamental physical quantities, find the dimensions of energy. [2021]

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Solution

Q.
If force [F], acceleration [A] and time [T] are chosen as the fundamental physical quantities, find the dimensions of energy. [2021]

1 $[F] [A^{- 1}] [T]$

2 $[F] [A] [T]$

3 $[F] [A] [T^{2}]$

4 $[F] [A] [T^{- 1}]$

Ans.
(3)

Let for energy, $[E] = F^{α} A^{β} T^{γ}$

or $[M^{1} L^{2} T^{- 2}]$ $=$ ${[M L T^{- 2}]}^{α}$ ${[L T^{- 2}]}^{β}$ ${[T]}^{γ}$

$M^{1} L^{2} T^{- 2} = M^{α} L^{α + β} T^{- 2 α - 2 β + γ}$

Comparing from both sides, $α = 1$

$α + β = 2 \Rightarrow β = 1$

$- 2 α - 2 β + γ = - 2 \Rightarrow γ = 2 ∴ Energy = [F] [A] [T^{2}]$

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