A uniform cylinder of steel of mass M, radius ’R’ is placed on frictionless bearings and set to rotate about its vertical axis with angular velocity ${\mathrm{\omega }}_0$ . After the cylinder has reached the specified state of rotation it is heated without any mechanical contact from temperature  T₀ to ${\mathrm{T}}_0+\mathrm{\Delta T}$ . If $\frac{\mathrm{\Delta I}}{\mathrm{I}}$ is the fractional change in moment of inertia of the cylinder and $\frac{\mathrm{\Delta \omega }}{{\mathrm{\omega }}_0}$ be the fractional change in angular velocity of cylinder and α be the coefficient of linear expansion then