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.