A uniform cylinder of steel of mass $M$, radius $R$ is placed on frictionless bearing and set to rotate about its axis with angular velocity ${\mathrm{\omega }}_0$. After the cylinder has reached the specified state of rotation, it is heated from temperature ${\mathrm{T}}_0$ to $\left({\mathrm{T}}_0+∆\mathrm{T}\right)$ without any mechanical contact. If $\frac{∆\mathrm{I}}{\mathrm{I}}$ is the fractional change in moment of inertia of the cylinder and $\frac{∆\mathrm{\omega }}{{\mathrm{\omega }}_0}$ be the fractional change is the angular velocity of the cylinder and $\mathrm{\alpha }$ be the coefficient of linear expansion, then