Question

$\huge\bold{Question}$
If force is
$f \: = \frac{ \alpha - {x}^{2} }{ \beta t}$
where t is time and xbis distance, then dimensions of
$a) \: \: \alpha \: \: is \: \: { l}^{2}$
$b) \: \beta \: \: is \: \: \: l$
$c) \: \alpha \: \: is \: \: ml {t}^{ - 2}$
$d) \: \beta \: \: is \: { m}^{ - 2} {t}^{4}$

​

Answer 1

Given :

Force equation is given by,

$\dag\:\underline{\boxed{\bf{\purple{F=\dfrac{\alpha-x^2}{\beta\cdot t}}}}}$

• x denotes distance
• t denotes time
• F denotes force
• α and β are constants

To Find :

Dimensions of α and β.

Solution :

❖ The principle of homogeneity of dimensions is used in checking the correctness of formulae, and also in the derivation of formulae

• Only like quantities having the same dimensions can be added to or substracted from each other.

[A] Dimensions of α :

$\sf:\implies\:[\alpha]=[x^2]$

$\sf:\implies\:[\alpha]=[(L^1)^2]$

$:\implies\:\underline{\boxed{\bf{\red{[\alpha]=[L^2]}}}}$

[B] Dimensions of β :

$\sf:\implies\:[\beta]=\dfrac{[L^2]}{[F][t]}$

$\sf:\implies\:[\beta]=\dfrac{[L^2]}{[M^1L^1T^{-2}][T^1]}$

$\sf:\implies\:[\beta]=\dfrac{[L^2]}{[M^1L^1T^{-1}]}$

$:\implies\:\underline{\boxed{\bf{\blue{[\beta]=[M^{-1}L^1T^1]}}}}$

∴ [A] is the correct answer!

Answer 2

Given :

Force equation is given by,

$\dag\:\underline{\boxed{\bf{\purple{F=\dfrac{\alpha-x^2}{\beta\cdot t}}}}}$

x denotes distance

t denotes time

F denotes force

α and β are constants

To Find :

Dimensions of α and β.

Solution :

❖ The principle of homogeneity of dimensions is used in checking the correctness of formulae, and also in the derivation of formulae

Only like quantities having the same dimensions can be added to or substracted from each other.

[A] Dimensions of α :

$\sf:\implies\:[\alpha]=[x^2]$

$\sf:\implies\:[\alpha]=[(L^1)^2]$

$:\implies\:\underline{\boxed{\bf{\red{[\alpha]=[L^2]}}}}$

[B] Dimensions of β :

$\sf:\implies\:[\beta]=\dfrac{[L^2]}{[F][t]}$

$\sf:\implies\:[\beta]=\dfrac{[L^2]}{[M^1L^1T^{-2}][T^1]}$

$\sf:\implies\:[\beta]=\dfrac{[L^2]}{[M^1L^1T^{-1}]}$

$:\implies\:\underline{\boxed{\bf{\blue{[\beta]=[M^{-1}L^1T^1]}}}}$

∴ [A] is the correct answer!