Question

Convert 1N into dyne using dimensional analysis

Answer 1

AnSwer :

To convert one system of unit into another system.

This is based on the fact that

forba given physical quantity, numerical value × unit = constant .i.e.,

• N₁U₁ = N₂U₂

Dimensional formula of froce is M¹L¹T¯²

M denotes Mass

L denotes length

T denotes time

1N = n dyne

n = N/dyne

${ \sf{\therefore n_2 = n_1 \bigg[\dfrac{M_1}{M_2}\bigg]^1 \bigg[\dfrac{L_1}{L_2}\bigg]^1 \bigg[\dfrac{T_1}{T_2}\bigg]^{-2}}}$

${: \implies{ \sf{\dfrac{ n_2}{n_1} = \bigg[\dfrac{M_1}{M_2}\bigg]^1 \bigg[\dfrac{L_1}{L_2}\bigg]^1 \bigg[\dfrac{T_1}{T_2}\bigg]^{-2}}}}$

${ : \implies{ \sf{n= \bigg[\dfrac{1 \: kg}{1 \: g}\bigg]^1 \bigg[\dfrac{1 \: m}{1 \: cm}\bigg]^1 \bigg[\dfrac{1 \: s}{1 \: s}\bigg]^{-2} \: \: \: \bigg( as \: \dfrac{n_2}{n_1} = n\bigg) }}}$

${ : \implies{ \sf{n= \bigg[\dfrac{1000 \: g}{1 \: g}\bigg]\bigg[\dfrac{100 \: cm}{1 \: cm}\bigg]^1 \bigg[\dfrac{1 \: s}{1 \: s}\bigg]^{-2} }}}$

${:\implies{ \sf{n=[1000 ] \times [100 ]^{ 1} \times [1] ^{ - 2} }}}$

${:\implies{ \sf{n= {10}^{5} }}}$

thus, 1newton = 10⁵ dyne

Alternative method :

${:\implies{ \sf{[F] = [MLT^{-2}]}}}$

${:\implies{ \sf{1 \: newton = 1(kg)(m)(s)^{ - 2} }}}$

${:\implies{ \sf{1 \: newton= 1 \times ( {10}^{3} g)( {10}^{2}cm)(s) ^{ - 2} }}}$

${:\implies{ \sf{1 \: newton= {10}^{5} ( g)(cm)(s) ^{ - 2} }}}$

${:\implies{ \sf{1 \: newton= {10}^{5} \: dyne }}}$

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Know more

some Applications of dimension :-

☄ Dimensional formula can be used to convert one system of unit into another system.

☄ Dimensional formula can be used to check the correctness of an equation.

☄ Dimensional formula can be used to find unit of a given quantity.

☄ Dimensional formula can be used to derived relationship among different physical quantities.

☄ Dimensional formula can be used to design our own new system of unit