Question

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

Answer 1

Hint : In order to the question, to find the dimensions of $a\,and\,b$ from the given equation $F = at + b{t^2}$ , first we will write the dimension of the parts of equation such that $at\,and\,b{t^2}$ that is equal to the Force and then we will separate $a\,and\,b$ at one side to find their dimensions.

Complete Step By Step Answer:
Given equation-
$F = at + b{t^2}$ …….(i)
Dimensions of $at\,and\,b{t^2}$ must be equal to the force.
Hence, $[F] = [{M^1}{L^1}{T^{ - 2}}]$ …….(ii)
$F$ from the equation(i) and (ii)-
$[at] = a[T] = [F] \\\ \therefore a[T] = [{M^1}{L^1}{T^{ - 2}}] \\\ \therefore [a] = [{M^1}{L^1}{T^{ - 3}}]$
And per $b{t^2} = b[{T^2}] = [{M^1}{L^1}{T^{ - 2}}]$
$\because [b] = [{M^1}{L^1}{T^{ - 4}}]$
Hence, the dimensions of $a\,and\,b$ are $[{M^1}{L^1}{T^{ - 3}}]\,\,and\,\,[{M^1}{L^1}{T^{ - 4}}]$ respectively.

Note :
Units are chosen by convention to convey magnitude or size, and dimensions are a representation of their essential nature. A sequence of occurrences, for example, has a set length of time. The dimension of duration is time.