How much energy is require to set a 0.304m phonograph record into rotation at 33.33 revolution per minute if the mass of the record is 0.11kg?

The mass of the turntable is not trivial, and probably more important for motor and drive design. However,

Thin disk moment of inertia= 1/2 m r^2

rotational energy gained= 1/2 I w^2

w=33.3*2PI*60

energy total= 1/2 (m *.304^2)(33.3*2PI*60)^2

you know m, and PI, compute in joules