If momentum (P),area (A),and time (T) are tak | vedclass

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(D) Let the dimensional formula for energy $(E)$ be expressed as $E = k P^a A^b T^c$,where $k$ is a dimensionless constant.Substituting the dimensions

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$[P A^{1/2} T^{-1}]$

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(D) Let the dimensional formula for energy $(E)$ be expressed as $E = k P^a A^b T^c$,where $k$ is a dimensionless constant.Substituting the dimensions of each quantity:$[M L^2 T^{-2}] = [M L T^{-1}]^a [L^2]^b [T]^c$$[M L^2 T^{-2}] = [M^a L^{a+2b} T^{-a+c}]$Comparing the powers of $M$,$L$,and $T$ on both sides:For $M$: $a = 1$For $L$: $a + 2b = 2 \implies 1 + 2b = 2 \implies 2b = 1 \implies b = 1/2$For $T$: $-a + c = -2 \implies -1 + c = -2 \implies c = -1$Substituting these values back into the expression,we get $E = [P^1 A^{1/2} T^{-1}]$.Thus,the correct option is $D$.

If momentum $(P)$,area $(A)$,and time $(T)$ are taken to be fundamental quantities,then the dimensional formula for energy is:

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