Question

In relation, $ F{\text{ }} = {\text{ }}a{\text{ }} + {\text{ }}bx $ , where F is the force, x is the distance. Calculate the dimensions of a and b.

Answer 1

Hint: The expressions or formulae which tell us how and which of the fundamental quantities are present in a physical quantity are known as the Dimensional Formula of the Physical Quantity. Dimensional formulae also help in deriving units from one system to another.

Complete step by step solution:
Given relation, $ F{\text{ }} = {\text{ }}a{\text{ }} + {\text{ }}bx $
Applying the principle of homogeneity of dimensions. We have,
Dimensions of a = dimensions of F
 $ = [{M^1}{L^1}{T^{ - 2}}].\; $
Dimension of box = dimensions of F
Dimensions of b = dimensions of $ \dfrac{f}{x} $ ​
= $ \dfrac{{[{M^1}{L^1}{T^{ - 2}}]}}{{[{L^1}]}} $
 $ = [{M^1}{L^0}{T^{ - 2}}] $
The conditions when we equivalent an actual amount with its dimensional formulae are called Dimensional Equations. The dimensional condition helps in communicating actual amounts regarding the base or principal amounts.
Assume there's an actual amount By which relies upon base amounts M (mass), L (Length), and T (Time) and their raised forces are a, b and c, at that point dimensional formulae of the actual amount [Y] can be communicated as
 $ \left[ Y \right] = \left[ {{M^a}{L^b}{T^b}} \right] $
Dimensional investigation is the act of checking relations between actual amounts by recognizing the elements of the actual amounts. These measurements are free of the mathematical products and constants and all the amounts on the planet can be communicated as an element of the essential measurements.
The dimensional examination is additionally used to reason the connection between at least two actual amounts. On the off chance that we know the level of reliance of an actual amount on another, that is how much one amount changes with the adjustment in another, we can utilize the rule of consistency of two articulations to discover the condition relating these two amounts. This can be seen all the more effectively through the accompanying delineation.

Note:
The possibility of estimation isn't restricted to real things. High-dimensional spaces as frequently as conceivable occur in number juggling and specialized orders. They may be limited spaces or arrangement spaces, for instance, in Lagrangian or Hamiltonian mechanics.