Question

A force F is given by $F=at+b{t}^2$ where, $t$ is the time. What are dimensions of $a$ and $b$ ?

• A. $[ML{T}^{-4}]$ and $[ML{T}^2]$
• B. $[ML{T}^{-2}]$ and $[ML{T}^0]$
• C. $[ML{T}^{-3}]$ and $[ML{T}^{-4}]$
• D. $[M{L}^2{T}^{-3}]$ and $[M{L}^3{T}^{-4}]$

Answer 1

Answer: (C)

Every equaiton relating physical quantities should be in dimensional balance.
The given equation is
$F=at+b{t}^2$
Every equation relating physical quantities should be in dimensional balance.
It means that the dimensions of the terms on both sides of the equation must be the same.
The reason for this is that only similar quantities can be equated.
Dimensions of force $F$ are
$F=\left[ML{T}^{-2}\right]$
[Comparing with first term of given expression we have]
$\left[ML{T}^{-2}\right]=[a][T]$
$\Rightarrow [a]=\left[ML{T}^{-3}\right]$
Similarly comparing with second term, we have
$\left[ML{T}^{-2}\right]=[b][T{]}^2$
$[b]=\left[ML{T}^{-1}\right]$
Hence, dimensions of $a$ and $b$ are $\left[ML{T}^{-3}\right]$ and $\left[ML{T}^{-4}\right]$ respectively.
The physical quantities separated by the symbols $+,-=,\lambda <$ etc. have the same dimensions.