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Sagot :
The dimension of acceleration is given by, [tex]m^{0} l^{1} t^{-2}[/tex], where m, l, t stand for mass, length and time respectively.
What is Dimension of any physical quantity?
A dimensional equation is created when a physical quantity is expressed as its dimensional formula. The basic numbers [L] for length, [M] for mass, and [T] for time are used to define dimensional formulas in these three dimensions.
The characteristics of a system are quantified using physical quantities. These physical quantities are given significance by the combination of a generally recognised unit and a numerical value. For instance, while describing a beam's length, we may state that it is 6 [numerical quantity] + meters [unit].
One crucial method for representing these systems in clear languages is through dimensional formulas. A few benefits come from using dimensional formulas to define derived units. It illustrates the representation of various physical quantities in terms of their fundamental units. It serves as a dimensional accuracy check for a derived quantity formula. It lessens errors that happen while translating between different systems' units (say from SI units to British units).
Derivation
We must obtain the formulas for displacement and velocity, two variables on which acceleration depends, in order to determine the dimensional formula for acceleration.
Dimensional Formula for Displacement
Change in position or movement is displacement, thus it is represneted by length.
[d] = [l]
This is also written as:
[d] = [[tex]m^{0} l^{1} t^{0}[/tex]]
This means that displacement has single dimension in length and no dimension in time and mass.
The dimension for displacement is [l].
Dimensional Formula for Velocity
Since velocity [v] is defined as a displacement with respect to time. This is represented as:
v = Displacement/Time
Since displacement [d] is given as [l] we have:
[v] = [l]/[t]
This can be further defined as:
[v] = [tex]lt^{-1}[/tex]
Finally concluded as:
[v] = [[tex]m^{0} l^{1} t^{-1}[/tex]]
The rate at which velocity changes over time is known as acceleration [a]. This is illustrative of:
[a] = Change in velocity/Change in time
Since we previously calculated velocity as [[tex]lt^{-1}[/tex]], acceleration is given as:
[a] =[tex][lt^{-1}]/[t][/tex]
This can be further condensed as follows:
[a]=[ [tex]lt^{-2}[/tex]]
This can also be stated as:
[a] = [[tex]m^{0} l^{1} t^{-2}[/tex]]
The dimensional formula for acceleration is displayed on the right side of the equation above. This dimensional equation states that acceleration has one length dimension, negative 2 of time dimensions, and no mass dimensions.
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