A force F is given by F=a t+b t^2 where t is time, what are...

A force $F$ is given by $F=a t+b t^{2}$ where $t$ is time, what are the dimension of $a$ and $b ?$
(A) $\left[M L T^{-1}, M L T^{0}\right]$,
(B) $\left[M L T^{-3}, M L^{2} T^{4}\right]$
(C) $\left[M L T^{-4}, M L T^{-1}\right]$
(D) $\left[M L^{-3} T, M L T^{-4}\right]$

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Key Concepts

Dimensional Analysis

Dimensional analysis is a technique used in physics and engineering to deduce the relationship between different physical quantities by comparing their fundamental dimensions such as mass, length, and time. It ensures that the equations describing physical phenomena are dimensionally consistent, meaning each term on both sides of the equality has the same dimensions.

Force

Force in physics is defined as the mass of an object multiplied by its acceleration, which gives it the dimensional formula MLT?². This provides a basis for analyzing equations involving force, allowing one to assign dimensions to other quantities or constants in the equation.

Equating Dimensions

Equating dimensions involves setting the dimensions of each term in a physical equation equal to the dimensions of the corresponding side of the equation. When unknown coefficients multiply variables with their own dimensions, this technique can determine the required dimensions of these coefficients to maintain dimensional consistency across the equation.

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