SI Unit of Force

As $ F = ma $

$ \displaystyle \therefore F=[M][L{{T}^{{-2}}}]=[{{M}^{1}}{{L}^{1}}{{T}^{{-2}}}]$

This is the dimensional formula of force.

The units of force are of two types: Absolute units and Gravitational units.

(i) Absolute units

(a) The absolute unit of force on SI is newton (represented by N).

One newton force is that much force which produces an acceleration of $ 1  m s^{-2} $ in a body of mass 1 kg.

$\displaystyle \begin{array}{l}\text{As }F=ma\\\therefore 1N=1\text{ kg}\times 1\text{ m}{{\text{s}}^{{\text{-2}}}}=1\text{ kgm}{{\text{s}}^{{\text{-2}}}}\end{array}$

(b) The absolute unit of force on c.g.s. system is dyne.

One dyne force is that much force which produces an acceleration of $ 1 cm s^{-2} $ in a body of mass one gram.

$ \displaystyle \begin{array}{l}\text{As }F=ma\\\therefore 1\text{ dyne}=1\text{ g}\times 1\text{ cm }{{\text{s}}^{{\text{-2}}}}=1\text{ g cm }{{\text{s}}^{{\text{-2}}}}\end{array}$

Relation between newton and dyne

$ \displaystyle \begin{array}{l}\text{As }1\text{N}=1\text{ kg}\times 1\text{ m}{{\text{s}}^{{\text{-2}}}}\\={{10}^{3}}\text{g}\times {{10}^{2}}\text{ cm }{{\text{s}}^{{-2}}}\\\therefore 1\text{N}={{10}^{5}}\text{dyne}\end{array}$

(ii) Gravitational units

(a) The gravitational unit of force on SI is 1 kilogram-weight (kg wt.), or 1 kilopond force (kg f). It is that much force which produces an acceleration of $ \displaystyle 9.8\text{ }\!\!~\!\!\text{ m}{{\text{s}}^{{-2}}}$ in a body of mass 1 kg.

$ \displaystyle 1\text{ kg wt}\text{. or }1\text{ kg f}$ $ \displaystyle =1\text{ kg}\times 9.8\text{ m}{{\text{s}}^{{-2}}}$ $ \displaystyle =9.8\text{N}$

(b) The gravitational unit of force on c.g.s. system is 1 gram weight (g wt.) or 1 gram force (gf). It is that much force which produces an acceleration of $ \displaystyle 980\text{ cm }{{\text{s}}^{{-2}}}$ in a body of mass 1 gram.

$ \displaystyle 1\text{ g wt}\text{.}=1\text{ g f}$ $ \displaystyle =1\text{ g}\times 980\text{ cm }{{\text{s}}^{{-2}}}$ $ \displaystyle =980\text{ dyne}$