1.Units, Dimensions and Measurement
hard

If force $(F),$ velocity $(V)$ and time $(T)$ are taken as fundamental units, then the dimensions of mass are 

A

$\left[ {FV{T^{ - 1}}} \right]$

B

$\;\left[ {FV{T^{ - 2}}} \right]$

C

$\;\left[ {F{V^{ - 1}}{T^{ - 1}}} \right]$

D

$\;\left[ {F{V^{ - 1}}T} \right]$

(AIPMT-2014)

Solution

Let, mass, m,  $\propto \,{F^a}{V^b}{T^e}$
or,  m$ = k{F^a}{V^b}{T^c}$  ………………..($i$)
Where k is a dimensionless constant and a, b and c are the exponents
Writing dimension on both sides we get

$\left[ {M{L^0}{T^0}} \right] = {\left[ {ML{T^{ – 2}}} \right]^a}{\left[ {L{T^{ – 1}}} \right]^b}{\left[ T \right]^c}$

$\left[ {M{L^0}{T^0}} \right] = \left[ {{M^a}{L^{a + b}}{T^{ – 2ab + c}}} \right]$
Applying the principle of homogeneity of

dimension we get

$a=1$     ………………($ii$)

$a + b = 0$         ……………………($iii$)

$- 2a – b + c = 0$     ………………..($iv$)
Solving eqns.,($ii$), ($iii$), and ($iv$), we get
$a = 1,b =  – 1,c = 1$
From eqn. $(i),\,\left[ m \right] = \left[ {F{V^{ – 1}}T} \right]$

Standard 11
Physics

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