Two discs of moments of inertia I₁ and I₂ about their respective axes, rotating with angular frequencies ${\mathrm{\omega }}_1$and ${\mathrm{\omega }}_2$  respectively, are brought into contact face to face with their axes of rotation coincident. The angular frequency of the composite disc will be

The initial angular momentum of the two discs
${\mathrm{L}}_1={\mathrm{I}}_1{\mathrm{\omega }}_1+{\mathrm{I}}_2{\mathrm{\omega }}_2$
The moment of inertia I_C. of the composite system will be
${\mathrm{I}}_{\mathrm{C}}={\mathrm{I}}_1+{\mathrm{I}}_2$
If ${\mathrm{\omega }}_{\mathrm{C}}$to" be the angular frequency of the composite system, then the final angular momentum L₂, is given by
${\mathrm{L}}_2={\mathrm{I}}_{\mathrm{c}}{\mathrm{\omega }}_{\mathrm{c}}=\left({\mathrm{I}}_1+{\mathrm{I}}_2\right){\mathrm{\omega }}_{\mathrm{c}}$
Applying the law of conservation of angular momentum, we have
$\left({\mathrm{I}}_1+{\mathrm{I}}_2\right){\mathrm{\omega }}_{\mathrm{C}}={\mathrm{I}}_{\mathrm{C}}{\mathrm{\omega }}_1+{\mathrm{I}}_2{\mathrm{\omega }}_2$
or ${\mathrm{\omega }}_{\mathrm{C}}=\frac{{\mathrm{I}}_1{\mathrm{\omega }}_1+{\mathrm{I}}_2{\mathrm{\omega }}_2}{{\mathrm{I}}_1+{\mathrm{I}}_2}$