Question

In the following equation x. t and F represent respectively displacement, time and force:
$F=a+bt+\frac{1}{c+dx}+Asin(\omega t+\varphi )$. The dimensional formula for Ad is

• A. $T^{-1}$
• B. $L^{-1}$
• C. $M^{-1}$
• D. $T{L}^{-1}$

Answer 1

The correct option is B $L^{-1}$

$F=a+bt+\frac{1}{c+dx}+Asin(\omega t+\varphi )$
The dimensions of A are same as that of force
$\therefore \left[A\right]=\left[F\right]=\left[ML{T}^{-2}\right]$
Also, the dimensions of $\frac{1}{c+dx}$ must be same as that of force
$\therefore \frac{1}{c+dx}=F\Rightarrow \left[\frac{1}{C}\right]=\left[F\right]$
$\therefore \left[c\right]=\left[M^{-1}{L}^{-2}{T}^2\right]$
Also, the dimension of dx is same as that of c
$\therefore \left[d\right]=\frac{\left[c\right]}{\left[x\right]}=\left[M^{-1}{L}^{-2}{T}^2\right]$
$\therefore \left[Ad\right]=\left[ML{T}^{-1}\right]\left[M^{-1}{L}^{-2}{T}^2\right]=\left[L^{-1}\right]$