Question 258: Units , Dimensions and Error Analysis

Question

A, B, C and D are four different physical quantities having different 
 dimensions. None of them is dimensionless. But we know that the equation AD = C ln (BD) holds true. Then which of the combination is
not
a meaningful quantity ?

Accepted Answer

Given,
AD = c ln (BD)
As log is dimensionless.
So,  [BD] = 1   $ \Rightarrow $  [B] = ${1 \over {\left[ D ight]}}$
And  [AD] = [C]
Now checking  options one by one
(a)   [B
2
C
2
] = [B
2
] [A
2
D
2
] = [A
2
] [B
2
D
2
] = [A
2
]
$ 	herefore $   This is meaningful.
(b)   $\left( {{{A - C} \over D}} ight)$ is not meaningful.
As dimension of A $ 
e $ dimension of C
Hence (A $-$ C)   is not possible.
(c)   $\left[ {{A \over B}} ight]$ = [AD] = [C]
$ 	herefore $   ${{A \over B}}$ $-$ C is meaningful.
(d)   $\left[ {{C \over {BD}}} ight]$ = ${{\left[ C ight]} \over {\left[ {BD} ight]}}$ = ${{\left[ C ight]} \over 1}$ = [C]
$\left[ {{{A{D^2}} \over C}} ight]$ = ${{\left[ {AD} ight]\left[ D ight]} \over {\left[ C ight]}}$ = ${{\left[ C ight]\left[ D ight]} \over {\left[ C ight]}}$ = [D]
As dimension  of C and D are not same so it is not meaning ful.

Answer

A
2
$-$ B
2
C
2

Answer

${{\left( {A - C} \right)} \over D}$

Answer

${A \over B} - C$

Answer

${C \over {BD}} - {{A{D^2}} \over C}$

Units , Dimensions and Error Analysis

A, B, C and D are four different physical quantities having different dimensions. None of them is dimensionless. But we know that the equation AD = C ln (BD) holds true. Then which of the combination is not a meaningful quantity ?

Choose the correct answer: