A unifom cylinder of steel of mass M radius R is placed on frictionless bearings and sct to rotate about its axis with angular velocity `omega_(0)` After the cylinder has reached the specified state of rotation, it is heated from temperature `T_(0)to (T_(0)+DeltaT)` without any mechanical contact. If (DeltaI)/I is the fractional change in moment of inertia of the cylinder and `(Deltaomega)/(omega_(0))` be the fractional change in the angular velocity of the cylinder and alpha be the coefficient of linear expansion, then

Options

A. `(DeltaI)/I=(2DeltaR)/R`
B. `(DeltaI)/I=(2Deltaomega)/(omega_(0))`
C. `(Deltaomega)/(omega_(0))=-2alphaDeltaT`
D. `(DeltaI)/I=-(2DeltaR)/R`

Answer

A, C

Explanation

`I=(MR^(2))/2` & M is constant Rightarrow `Ialpha R^(2)`
:.`(DeltaI)/I=2((DeltaR)/R)=2alpha(DeltaT)` on heating also no external torque present :.L=Iw=constant
w `alpha 1/I`
:.`(Deltaw)/w=-(DeltaI)/I=-2((DeltaR)/R)=-2alpha(DeltaT)`