First slide

Dimensions of physical quantities

Question

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

Easy

A

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

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

$[P^{2} A T^{- 2}]$
D

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

Solution

Let Energy be expressed as powers of momentum(P), Area (A) and time (T), then
$\begin{array}{l} [E] = {[P]}^{a} {[A]}^{b} {[T]}^{c} \\ [M^{1} L^{2} T^{- 2}] = {[M^{1} L^{1} T^{- 1}]}^{a} {[L^{2}]}^{b} {[T]}^{c} \end{array}$
comparing powers on both sides,
1 = a
2 = a + 2b
-2 = -a + c
$∴$ b = 1/2 and c = -1

$[E] = P A^{\frac{1}{2}} T^{- 1}$

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