If force [F], acceleration [A] and time [T] are chosen as fundamental physical quantities. dimension

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1.Units, Dimensions and Measurement

medium

If force $[F],$ acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical quantities. Find the dimensions of energy.

$[\mathrm{F}][\mathrm{A}][\mathrm{T}]$

$[\mathrm{F}][\mathrm{A}]\left[\mathrm{T}^{2}\right]$

$[F][\mathrm{A}]\left[\mathrm{T}^{-1}\right]$

$[\mathrm{F}]\left[\mathrm{A}^{-1}\right][\mathrm{T}]$

(NEET-2021)

Solution

$\mathrm{E} \propto \mathrm{F}^{\mathrm{a}} \mathrm{A}^{\mathrm{b}} \mathrm{T}^{\mathrm{c}}$

$\left[\mathrm{M}^{1} \mathrm{~L}^{2} \mathrm{~T}^{2}\right] \propto\left[\mathrm{M}^{1} \mathrm{~L}^{1} \mathrm{~T}^{-2}\right]^{\mathrm{a}}\left[\mathrm{LT}^{-2}\right]^{\mathrm{b}}[\mathrm{T}]^{\mathrm{c}}$

$\mathrm{a}=1$

$\mathrm{a}+\mathrm{b}=2 \Rightarrow \mathrm{b}=1$

$-2 \mathrm{a}-2 \mathrm{~b}+\mathrm{c}=-2$

$\Rightarrow \mathrm{c}=2$

$\mathrm{a}=1 \mathrm{~b}=1 \mathrm{c}=2$

$\mathrm{E} \propto[\mathrm{F}][\mathrm{A}]\left[\mathrm{T}^{2}\right]$

Standard 11

Physics

The $SI$ unit of energy is $J=k g\, m^{2} \,s^{-2} ;$ that of speed $v$ is $m s^{-1}$ and of acceleration $a$ is $m s ^{-2} .$ Which of the formulae for kinetic energy $(K)$ given below can you rule out on the basis of dimensional arguments ( $m$ stands for the mass of the body ):

$(a)$ $K=m^{2} v^{3}$

$(b)$ $K=(1 / 2) m v^{2}$

$(c)$ $K=m a$

$(d)$ $K=(3 / 16) m v^{2}$

$(e)$ $K=(1 / 2) m v^{2}+m a$

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(KVPY-2017)

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If force $[F],$ acceleration $[A]$ and time $[T]$ are chosen as the fundamental physical quantities. Find the dimensions of energy.

Solution

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