First slide

Dimensions of physical quantities

Question

$\int \frac{d t}{\sqrt{2 a t - t^{2}}} = a^{x} \sin^{- 1} [\frac{t}{a} - 1]$ The value of x is

Moderate

A

1
B

-1
C

0
D

2

Solution

$The quantity \frac{t}{a} - 1 is dimensionless i.e., [a] = [t]$
$\begin{array}{l} ∴ [\sqrt{2 a t - t^{2}}] = [t] \\ or [\frac{d t}{\sqrt{2 a t} - t^{2}}] = [\frac{t}{t}] = [M^{0} L^{0} T^{0}] \end{array}$
$i.e. a^{x} should also be dimensionless or x = 0 .$

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