The frictional moment M acting on a rotating turbine disk and its shaft is given by

The frictional moment M_ƒ acting on a rotating turbine disk and its shaft is given by M_ƒ = kw² where w is the angular velocity of the turbine. If the source of power is cut off while the turbine is running with an angular velocity w₀, determine the time t for the speed of the turbine to drop to one-half of its initial value. The moment of inertia of the turbine disk and shaft is I.