Question 230: 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: