Question:
If $force (F)$, $velocity (V)$ and $time (T)$ are taken as fundamental units, then the dimensions of mass are
If $force (F)$, $velocity (V)$ and $time (T)$ are taken as fundamental units, then the dimensions of mass are
Updated On: Jul 12, 2022
- $[FVT^{-1}]$
- $[FVT^{-2}]$
- $[FV^{-1}T^{-1}]$
- $[FV^{-1}T]$
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The Correct Option is D
Solution and Explanation
Let mass $m \propto F^aV^bT^c$
or $m=kF^aV^bT^c \ldots (i)$
where $k$ is a dimensionless constant and $a$, $b$ and $c$ are the exponents.
Writing dimensions on both sides, we get
$[ML^0T^0]=[MLT^{-2}]^a[LT^{-1}]^b[T]^c$
$[ML^0T^0]=[M^aL^{a+b}T^{-2a-b+c}]$
Applying the principle of homogeneity of dimensions, we get
$a=1 \ldots (ii)$
$a+b=0 \ldots (iii)$
$-2a-b+c=0 \ldots (iv)$
Solving eqns. $(ii)$, $(iii)$ and $(iv)$, we get
$a= 1$,
$b = -1$,
$c = 1$
From eqn. $(i)$, $[m] = [FV^{-1}T]$
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Concepts Used:
Units and Measurement
Unit:
A unit of a physical quantity is an arbitrarily chosen standard that is broadly acknowledged by the society and in terms of which other quantities of similar nature may be measured.
Measurement:
The process of measurement is basically a comparison process. To measure a physical quantity, we have to find out how many times a standard amount of that physical quantity is present in the quantity being measured. The number thus obtained is known as the magnitude and the standard chosen is called the unit of the physical quantity.
Read More: Fundamental and Derived Units of Measurement
System of Units:
- CGS system
- FPS system
- MKS system
- SI units
Types of Units:
Fundamental Units -
The units defined for the fundamental quantities are called fundamental units.
Derived Units -
The units of all other physical quantities which are derived from the fundamental units are called the derived units.