If momentum (\(\mathrm{p}\)), area (\(\mathrm{A}\)), and time (\(\mathrm{T}\)) are taken to be fundamental quantities, then energy has the dimensional formula:
1. \( {\left[\mathrm{pA}^{-1} \mathrm{~T}^1\right]} \)
2. \( {\left[\mathrm{p}^2 \mathrm{AT}\right]} \)
3. \( {\left[\mathrm{pA}^{-1 / 2} \mathrm{~T}\right]} \)
4. \( {\left[\mathrm{pA}^{1 / 2} \mathrm{~T}^{-1}\right]}\)
 

(4) Hint: Use the dimensional analysis to find the correct relation.

Step 1: Define a relation for energy.

Given, fundamental quantities are momentum (p), area (A), and time (T).

We can write energy E as

EpaAbTcE=kpaAATc

where k is dimensionless constant ot proportionality.

Step 2: Put all the dimensions.

Dimensions of E=[E]=[ML2T2] and [p]=[MLT1]
[A]=[L2]
[T]=[T]
[E]=[K][p]a[A]b[T]c

Putting all the dimensions, we get

ML2T2=[MLT1]a[L2]b[T]c=MaL2b+aTa+c

Step 3: Compare the dimensions.

By the principle of homogeneity of dimensions,

a=1, 2b+a=2
 2b+1=2
 b=1/2a+c=2
 c=2+a=2+1=1
Hence, E=pA1/2T1