First slide

Dimensions of physical quantities

Question

The velocity $v$ of a particle depends upon the time t according to the equation, $v = a + b t + \frac{c}{d + t}$ . Write the dimensions of $a, b, c a n d d$ .

Moderate

A

$[a] = [v], [b] = [{LT}^{2}], [c] = [L], [d] = [T]$
B

$[a] = [v], [b] = [{LT}^{- 2}], [c] = [L], [d] = [T]$
C

$[a] = [v], [b] = [{LT}^{- 2}], [c] = [L], [d] = [L]$
D

$[a] = [v], [b] = [{LT}^{- 3}], [c] = [L], [d] = [T]$

Solution

From principle of homogeneity,
$[a] = [v] o r [a] = [{LT}^{- 1}] [b t] = [v] o r [b] = \frac{[v]}{[t]} = \frac{[{LT}^{- 1}]}{[T]}$
or,
$[b] = [{LT}^{- 2}]$
Similarly,
$[d] = [t] = [T]$
Further,
$\frac{[c]}{[d + t]} = [v] o r [c] = [v] [d + t]$
or,
$[c] = [{LT}^{- 1}] [T] o r [c] = [L]$

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