The uniform disk of radius R and thickness t is attached to the thin shaft of radius r, le...

The uniform disk of radius R and thickness t is attached to the thin shaft of radius r, length L, and negligible mass. The end A of the shaft is fixed. From mechanics of materials, it can be shown that if a torque Mz is applied to the free end of the shaft, then it can be related to the twist angle θ via

where G is the shear modulus of elasticity of the shaft and is the polar moment of inertia of the cross-sectional area of the shaft. Letting ρ be the mass density of the disk and using the given relationship between Mz and θ, determine the natural frequency of vibration of the disk in terms of the given dimensions and material properties when it is given a small angular displacement θ in the plane of the disk.

Figure P9.8